{"id":3364,"date":"2024-01-30T12:15:59","date_gmt":"2024-01-30T06:15:59","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3364"},"modified":"2024-11-05T16:39:29","modified_gmt":"2024-11-05T10:39:29","slug":"algebraic-fractions","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/algebraic-fractions\/","title":{"rendered":"\u09ae\u09c2\u09b2\u09a6 \u09ac\u09c0\u099c\u0997\u09a3\u09bf\u09a4\u09c0\u09af\u09bc \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6\u09c7\u09b0 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3; \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c, \u098f\u09b0 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09bf\u09a4 \u09ae\u09c2\u09b2 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af, \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09c7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2"},"content":{"rendered":"

\u09ae\u09c2\u09b2\u09a6 \u09ac\u09c0\u099c\u0997\u09a3\u09bf\u09a4\u09c0\u09af\u09bc \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6\u09c7\u09b0 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 <\/b>(Integration of Rational Algebraic Fractions):<\/b><\/h2>\n

\u0995\u09cb\u09a8\u09cb \u09ae\u09c2\u09b2\u09a6 \u09ac\u09c0\u099c\u0997\u09a3\u09bf\u09a4\u09c0\u09af\u09bc \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6\u09c7\u09b0 \u09af\u09cb\u0997\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09be\u09b0 \u099c\u09a8\u09cd\u09af \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6\u099f\u09bf\u0995\u09c7 \u0986\u0982\u09b6\u09bf\u0995 \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6\u09c7 \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3 \u0995\u09b0\u09c7 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995 \u0985\u0982\u09b6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09af\u09cb\u099c\u09bf\u09a4 \u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

\u09a8\u09bf\u09af\u09bc\u09ae (Rule)<\/b>: \\frac{a x^{2}+b x+c}{(x-a)(x-\\beta)^{2}\\left(p x^{2}+\\gamma\\right)} \\equiv \\frac{A}{x-\\alpha}+\\frac{B}{(x-\\beta)^{2}}+\\frac{C}{x-\\beta}+\\frac{D x+E}{p x^{2}+\\gamma}<\/span><\/span><\/p>\n

 <\/p>\n

\u0985\u09ad\u09bf\u099c\u09cd\u099e\u09a4\u09be\u09b2\u09ac\u09cd\u09a7 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf (Thumb Rule Method):<\/b><\/p>\n

\u09af\u09a6\u09bf \u0995\u09cb\u09a8\u09cb \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6\u09c7\u09b0 \u09b9\u09b0\u09c7\u09b0 \u0989\u09ce\u09aa\u09be\u09a6\u0995\u09b8\u09ae\u09c2\u09b9 \u098f\u0995\u0998\u09be\u09a4 \u09b9\u09af\u09bc \u098f\u09ac\u0982 \u0995\u09cb\u09a8\u099f\u09bf\u09b0\u0987 \u09aa\u09c1\u09a8\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4\u09bf \u09a8\u09be \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u09a4\u09be\u09b0 \u0986\u0982\u09b6\u09bf\u0995 \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6 \u0985\u09a4\u09bf \u09b8\u09b9\u099c\u09c7 \u09a8\u09bf\u09ae\u09cd\u09a8\u09c7\u09b0 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 (\u0985\u09ad\u09bf\u099c\u09cd\u099e\u09a4\u09be\u09b2\u09ac\u09cd\u09a7 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf) \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc:<\/span><\/p>\n

\\frac{x+c}{(x-a)(x+b)}=\\frac{a+c}{(x-a)(a+b)}+\\frac{-b+c}{(-b-a)(x+b)}<\/span>,\u00a0<\/span><\/p>\n\\frac{x^{2}-c}{\\left(x^{2}+a\\right)\\left(x^{2}-b\\right)}=\\frac{-a-c}{\\left(x^{2}+a\\right)(-a-b)}+\\frac{b-c}{(b+a)\\left(x^{2}-b\\right)}<\/span>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09ec: <\/b>\\int \\frac{x^{3}-2 x+3}{x^{2}+x-2} d x<\/span>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

(Example \u2013 6: Determine <\/b>\\int \\frac{x^{3}-2 x+3}{x^{2}+x-2} d x<\/span>)<\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n\\frac{x^{3}-2 x+3}{x^{2}+x-2}=\\frac{x\\left(x^{2}+x-2\\right)-1\\left(x^{2}+x-2\\right)+x+1}{x^{2}+x-2}<\/span>\n

 <\/p>\n=\\frac{x\\left(x^{2}+x-2\\right)}{x^{2}+x-2}-\\frac{x^{2}+x-2}{x^{2}+x-2}+\\frac{x+1}{x^{2}+x-2}=x+1+\\frac{x+1}{(x+2)(x-1)}<\/span>\n

 <\/p>\n=x+1+\\frac{-2+1}{(x+2)(-2-1)}+\\frac{1+1}{(1+2)(x-1)}=x+1+\\frac{1}{3(x+2)}+\\frac{2}{3(x-1)}<\/span>\n

 <\/p>\n\\therefore \\int \\frac{x^{3}-2 x+3}{x^{2}+x-2} d x=\\int\\left\\{x+1+\\frac{1}{3(x+2)}+\\frac{2}{3(x-1)}\\right\\} d x<\/span>\n

 <\/p>\n

=\\frac{x^{2}}{2}+x+\\frac{1}{3} \\ln |x+2|+\\frac{2}{3} \\ln |x-1|+c<\/span>\u00a0 \u00a0 (Ans)<\/strong><\/p>\n

 <\/p>\n

\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c (The Definite Integral)<\/b>:<\/b><\/h2>\n

\u09af\u09a6\u09bf <\/span>[a,\u00a0b]<\/span> \u09ac\u09a6\u09cd\u09a7 \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 <\/span>f(x)<\/span> \u09ab\u09be\u0982\u09b6\u09a8 \u09b8\u09c0\u09ae\u09be\u09ac\u09a6\u09cd\u09a7 (Bounded) \u09b9\u09af\u09bc \u098f\u09ac\u0982 <\/span>[a,\u00a0b]<\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u0995\u09c7 <\/span>n<\/span> \u09b8\u0982\u0996\u09cd\u09af\u0995 \u0989\u09aa\u09ac\u09cd\u09af\u09ac\u09a7\u09bf \\delta_{r}=\\left[x_{r-1}, x_{r}\\right], r=1,2,3,4 \\ldots \\ldots n<\/span> <\/span>\u098f \u098f\u09b0\u09c2\u09aa\u09c7 \u09ac\u09bf\u09ad\u0995\u09cd\u09a4 \u0995\u09b0\u09be \u09b9\u09af\u09bc \u09af\u09c7, \u09b8\u09b0\u09cd\u09ac\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09be \u09a6\u09c0\u09b0\u09cd\u0998 \u0989\u09aa\u09ac\u09cd\u09af\u09ac\u09a7\u09bf <\/span>\u03b4\u21920<\/span> \u09b9\u09df \u098f\u09ac\u0982 \\xi_{r} \\in \\delta_{r}<\/span> <\/span>\u098f\u09b0 \u099c\u09a8\u09cd\u09af \\lim _{\\delta \\rightarrow 0} \\sum f\\left(\\xi_{r}\\right) \\delta_{r}<\/span><\/span>\u00a0\u098f\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u098f\u0995\u099f\u09bf \u09ae\u09be\u09a4\u09cd\u09b0 \u09b8\u09b8\u09c0\u09ae \u09ae\u09be\u09a8 \u09a5\u09be\u0995\u09c7, \u09a4\u09ac\u09c7 \u09b8\u09c7\u0987 \u09ae\u09be\u09a8\u0995\u09c7 \u09a8\u09bf\u09ae\u09cd\u09a8\u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4 <\/span>a<\/span> \u09b9\u09a4\u09c7 \u098a\u09b0\u09cd\u09a7\u09cd\u09ac\u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4 <\/span>b<\/span> \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4 <\/span>f(x)<\/span> \u098f\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09ac\u09b2\u09be \u09b9\u09af\u09bc \u098f\u09ac\u0982 \u0989\u09b9\u09be\u0995\u09c7\\int_{a}^{b} f(x) d x<\/span> <\/span>\u00a0\u09aa\u09cd\u09b0\u09a4\u09c0\u0995 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \\lim _{\\delta \\rightarrow 0} \\sum f\\left(\\xi_{r}\\right) \\delta_{r}=\\int_{a}^{b} f(x) d x<\/span><\/span><\/p>\n

\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c\u09c7\u09b0 \u09ae\u09be\u09a8<\/b>: \u09af\u09a6\u09bf <\/span>x<\/span> \u0995\u09c7 \u099a\u09b2\u09b0\u09be\u09b6\u09bf \u09a7\u09b0\u09c7 <\/span>[<\/span>a,\u00a0b]<\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 <\/span>f(x)<\/span> \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09af\u09cb\u0997\u099c <\/span>F(x)<\/span> \u09b9\u09af\u09bc \u0985\u09b0\u09cd\u09a5\u09be\u09ce<\/span> \\int_{a}^{b} f(x) d x<\/span> <\/span>= F(x) <\/span>\u09b9\u09af\u09bc \u09a4\u09ac\u09c7 <\/span>F(b) \u2013 F(a)<\/span> \u0995\u09c7 <\/span>f(x)<\/span> \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c\u09c7\u09b0 \u09ae\u09be\u09a8 \u09ac\u09b2\u09be \u09b9\u09af\u09bc \u098f\u09ac\u0982 \u098f\u0995\u09c7 \\int_{a}^{b} f(x) d x<\/span><\/span>\u00a0\u09aa\u09cd\u09b0\u09a4\u09c0\u0995 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09a8\u09bf\u099a\u09c7\u09b0 \u09aa\u09a6\u0995\u09cd\u09b7\u09c7\u09aa\u0997\u09c1\u09b2\u09bf \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u099c\u09a8\u0964<\/span><\/p>\n

\\int_{a}^{b} f(x) d x=[F(x)]_{a}^{b}=F(a)-F(b),<\/span><\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span>\u222b f(x)dx = F(x)<\/span><\/p>\n

 <\/p>\n

\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09bf\u09a4 \u09ae\u09c2\u09b2 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af (Primary theorems related to definite Integral):<\/b><\/h2>\n

\u0985\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09ae\u09c2\u09b2\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \\int_{a}^{b} f(x) d x<\/span><\/span>\u00a0<\/span>\u00a0<\/span>\u09af\u09be\u09b0 \u098a\u09b0\u09cd\u09a7\u09cd\u09ac\u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4 \u099a\u09b2\u09b0\u09be\u09b6\u09bf\u0964 <\/span>f(x)<\/span> \u0985\u09ac\u09bf\u099a\u09cd\u099b\u09bf\u09a8\u09cd\u09a8 \u09ab\u09be\u0982\u09b6\u09a8 \u09b9\u09b2\u09c7 <\/span>F(<\/span>x) <\/span>=\\int_{a}^{b} f(x) d x<\/span>\u00a0<\/span>\u00a0\u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3\u09af\u09cb\u0997\u09cd\u09af \u098f\u09ac\u0982<\/span>\u00a0<\/span>F<\/span>‘(<\/span>x) <\/span>= f(<\/span>x)<\/span>.G(x)<\/span> \u09af\u09a6\u09bf <\/span>f(x)<\/span> \u098f\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09aa\u09cd\u09b0\u09a4\u09bf\u0985\u09a8\u09cd\u09a4\u09b0\u099c \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \\int_{a}^{b} f(x) d x=G(b)-G(a)<\/span>;<\/span>\u00a0\u098f\u0987 \u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09bf\u09a4 \u09ae\u09c2\u09b2 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af \u09a8\u09be\u09ae\u09c7 \u09aa\u09b0\u09bf\u099a\u09bf\u09a4\u0964 \u098f\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09a6\u09c1\u0987\u099f\u09bf \u09aa\u09cd\u09b0\u09a4\u09bf\u0985\u09a8\u09cd\u09a4\u09b0\u099c\u09c7\u09b0 \u09aa\u09be\u09b0\u09cd\u09a5\u0995\u09cd\u09af \u09a7\u09cd\u09b0\u09c1\u09ac \u09ab\u09be\u0982\u09b6\u09a8 \u09ac\u09bf\u09a7\u09be\u09af\u09bc, \u098f\u0995\u09a6\u09bf\u0995\u09c7 <\/span>G(<\/span>b) <\/span>– G(<\/span>a)<\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09aa\u09cd\u09b0\u09a4\u09bf\u0985\u09a8\u09cd\u09a4\u09b0\u099c\u09c7\u09b0 \u099a\u09af\u09bc\u09a8\u09c7\u09b0 \u0989\u09aa\u09b0 \u09a8\u09bf\u09b0\u09cd\u09ad\u09b0 \u0995\u09b0\u09c7 \u09a8\u09be; \u0985\u09a8\u09cd\u09af\u09a6\u09bf\u0995\u09c7 <\/span>G(x)<\/span> \u09af\u09a6\u09bf <\/span>f(x)<\/span> \u098f\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09aa\u09cd\u09b0\u09a4\u09bf\u0985\u09a8\u09cd\u09a4\u09b0\u099c \u09b9\u09af\u09bc \u09a4\u09ac\u09c7 <\/span>G(<\/span>x) <\/span>= f(<\/span>x) <\/span>+ c,<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7<\/span>\u00a0<\/span>c<\/span> \u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u09b8\u0982\u0996\u09cd\u09af\u09be\u0964<\/span><\/p>\n\\int_{a}^{b} f(x) d x=[F(x)+c]_{a}^{b}=\\{F(b)+c\\}-\\{F(a)+c\\}=F(b)+c-F(a)-c=F(b)-F(a)<\/span>\n

\u098f\u09b0 \u09ab\u09b2\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09aa\u09cd\u09b0\u09b8\u0999\u09cd\u0997\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf\u0985\u09a8\u09cd\u09a4\u09b0\u099c \u098f\u09ac\u0982 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09b8\u09ae\u09be\u09b0\u09cd\u09a5\u0995 \u09ac\u09b2\u09c7 \u09ac\u09bf\u09ac\u09c7\u099a\u09a8\u09be \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964 \u09a4\u09be\u0987 \u0995\u09cd\u09af\u09be\u09b2\u0995\u09c1\u09b2\u09be\u09b8\u09c7 \u0985\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09ac\u09b2\u09a4\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf\u0985\u09a8\u09cd\u09a4\u09b0\u099c\u0995\u09c7\u0987 \u09ac\u09c1\u099d\u09be\u09a8\u09cb \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af<\/b>:<\/span> \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c\u09c7 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09a5\u09be\u0995\u09c7 \u09a8\u09be\u0964<\/span><\/p>\n

\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c\u09c7\u09b0 \u0995\u09bf\u099b\u09c1 \u09a7\u09b0\u09cd\u09ae (Few laws of definite Integral theorem):<\/b><\/p>\n

    \n
  1. \\int_{a}^{b} f(x) d x=\\int_{a}^{b} f(z) d z<\/span><\/li>\n
  2. \\text { 2. } \\int_{a}^{b} f(x) d x=-\\int_{b}^{a} f(x) d x<\/span><\/li>\n
  3. \\int_{0}^{a} f(x) d x=\\int_{0}^{a} f(a-x) d x<\/span><\/li>\n<\/ol>\n

     <\/p>\n

    \u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e7: <\/b>\\int_{0}^{\\pi \/ 4} \\frac{1}{1+\\sin x} d x<\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

    (Example \u2013 1: <\/b>\\int_{0}^{\\pi \/ 4} \\frac{1}{1+\\sin x} d x<\/span>\u00a0<\/span>determine its value)<\/b><\/p>\n

    \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n

    \\int_{0}^{\\pi \/ 4} \\frac{1}{1+\\sin x} d x=\\int_{0}^{\\pi \/ 4} \\frac{1-\\sin x}{(1+\\sin x)(1-\\sin x)} d x=\\int_{0}^{\\pi \/ 4} \\frac{1-\\sin x}{1-\\sin ^{2} x} d x<\/span><\/span><\/p>\n

    =\\int_{0}^{\\pi \/ 4} \\frac{1-\\sin x}{\\cos ^{2} x} d x<\/span><\/span><\/p>\n

    =\\int_{0}^{\\pi \/ 4}\\left(\\frac{1}{\\cos ^{2} x}-\\frac{\\sin x}{\\cos ^{2} x}\\right) d x<\/span><\/span>\u00a0<\/span><\/p>\n

    =\\int_{0}^{\\pi \/ 4}\\left(\\sec ^{2}-\\sec x \\tan x\\right) d x<\/span><\/span>\u00a0<\/span><\/p>\n

    =[\\tan x-\\sec x]_{0}^{\\pi \/ 4}<\/span><\/span><\/p>\n

    =\\left(\\tan \\frac{\\pi}{4}-\\sec \\frac{\\pi}{4}\\right)-(\\tan 0-\\sec 0)<\/span><\/span><\/p>\n

    =(1-\\sqrt{2})-(0-1)<\/span><\/span><\/p>\n

    =2-\\sqrt{2}<\/span><\/span><\/p>\n

     <\/p>\n

    \u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e8: <\/b>\\int_{0}^{\\pi \/ 2} \\cos ^{3} x \\sqrt{\\sin x} d x<\/span>\u00a0\u098f\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

    (Example \u2013 2: \\int_{0}^{\\pi \/ 2} \\cos ^{3} x \\sqrt{\\sin x} d x<\/span><\/b>\u00a0determine its value)<\/b><\/p>\n

    \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n\\int_{0}^{\\pi \/ 2} \\cos ^{3} x \\sqrt{\\sin x} d x=\\int_{0}^{\\pi \/ 2} \\cos ^{2} x \\cos x \\sqrt{\\sin x} d x<\/span>\n

     <\/p>\n

    =\\int_{0}^{\\pi \/ 2}\\left(1-\\sin ^{2} x\\right) \\sqrt{\\sin x} \\cos x d x<\/span><\/p>\n

     <\/p>\n

    \u09a7\u09b0\u09bf, \\sin x=z \\therefore \\cos x d x=d z<\/span><\/span><\/p>\nx=0\u00a0 \u09b9\u09b2\u09c7, z=\\sin 0=0 ; x=\\frac{\\pi}{2}\u00a0 \u09b9\u09b2\u09c7, z=\\sin \\frac{\\pi}{2}=1<\/span>\n

     <\/p>\n\\therefore \\int_{0}^{\\pi \/ 2} \\cos ^{3} x \\sqrt{\\sin x} d x=\\int_{0}^{1}\\left(1-z^{2}\\right) \\sqrt{z} d z<\/span>\n

     <\/p>\n=\\int_{0}^{1}\\left(\\sqrt{z}-z^{5 \/ 2}\\right) d z<\/span>\n

     <\/p>\n=\\left[\\frac{z^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}-\\frac{z^{\\frac{5}{2}+1}}{\\frac{5}{2}+1}\\right]_{0}^{1}<\/span>\n

     <\/p>\n=\\left[\\frac{2 z^{\\frac{3}{2}}}{3}-\\frac{2 z^{\\frac{7}{2}}}{7}\\right]_{0}^{1}<\/span>\n

     <\/p>\n=\\left[\\frac{2(1)^{\\frac{3}{2}}}{3}-\\frac{2(1)^{\\frac{7}{2}}}{7}\\right]-0<\/span>\n

     <\/p>\n=\\frac{2}{3}-\\frac{2}{7}=\\frac{8}{21}<\/span>\n

     <\/p>\n

    \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09c7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 (Determining area using definite integral)<\/b><\/h2>\n

    (a,\u00a0b)<\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09bf\u09a4\u09c7 <\/span>y = f(x)<\/span> \u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf \u0985\u09ac\u09bf\u099a\u09cd\u099b\u09bf\u09a8\u09cd\u09a8 \u09b9\u09b2\u09c7 <\/span>y = f(<\/span>x)<\/span>,\u00a0 x<\/span>-\u0985\u0995\u09cd\u09b7 \u098f\u09ac\u0982<\/span>\u00a0<\/span>x = a\u00a0<\/span> \u0993\u00a0 <\/span>x = b<\/span> \u0995\u09cb\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0986\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 =<\/span> \\int_{a}^{b} f(x) d x<\/span><\/span><\/p>\n

    <\/h1>\n

    \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,<\/span>\u00a0<\/span>x = a<\/span> \u0993 <\/span>x = b<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0995\u09cb\u099f\u09bf \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>AB<\/span> \u0993<\/span>\u00a0<\/span>DC, y = f(x)<\/span> \u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be\u0995\u09c7<\/span>\u00a0<\/span>A<\/span> \u0993 <\/span>D<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7\u0964<\/span>\u00a0<\/span>ABCD<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0986\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

    \u09a7\u09b0\u09bf,<\/span>\u00a0<\/span>AD<\/span> \u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 <\/span>P(x,\u00a0y)<\/span> \u098f\u09ac\u0982<\/span> Q(x+\\delta x, y+\\delta y)<\/span> <\/span>\u09a6\u09c1\u0987\u099f\u09bf \u09a8\u09bf\u0995\u099f\u09ac\u09b0\u09cd\u09a4\u09c0 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \\delta x \\rightarrow 0<\/span><\/span> \u09b9\u09b2\u09c7, \\delta y \\rightarrow 0<\/span>.<\/span>\u00a0x<\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0<\/span>\u00a0<\/span>PM<\/span> \u0993<\/span>\u00a0<\/span>QN<\/span> \u09b2\u09ae\u09cd\u09ac \u099f\u09be\u09a8\u09bf\u0964<\/span><\/p>\n

    QN<\/span> \u098f\u09b0 \u0989\u09aa\u09b0<\/span>\u00a0<\/span>PR<\/span> \u098f\u09ac\u0982<\/span>\u00a0<\/span>MP<\/span> \u098f\u09b0 \u09ac\u09b0\u09cd\u09a7\u09bf\u09a4\u09be\u0982\u09b6\u09c7\u09b0 \u0989\u09aa\u09b0<\/span>\u00a0<\/span>QS<\/span> \u09b2\u09ae\u09cd\u09ac \u099f\u09be\u09a8\u09bf\u0964<\/span><\/p>\n\\therefore O M=x, O N=x+\\delta x, P M=y, Q N=y+\\delta y<\/span>\n

    \u00a0<\/span>\u09b8\u09c1\u09a4\u09b0\u09be\u0982, M N=(x+\\delta x)-x=\\delta x<\/span><\/span><\/p>\n

    \u099a\u09bf\u09a4\u09cd\u09b0 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span>\u00a0<\/span>MNRP <\/span>\u0986\u09af\u09bc\u09a4\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 = y \\delta x<\/span><\/span>\u00a0\u098f\u09ac\u0982<\/span>\u00a0<\/span>MNQS<\/span> \u0986\u09af\u09bc\u09a4\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 =(y+\\delta y) \\delta x<\/span><\/span><\/p>\n

    ABMP<\/span> \u098f\u09ac\u0982<\/span>\u00a0<\/span>ABNQ<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>A<\/span> \u098f\u09ac\u0982 A+\\delta A<\/span><\/span>\u00a0\u09b9\u09b2\u09c7,<\/span>\u00a0<\/span>PMNQ<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 =\\delta A<\/span><\/span><\/p>\n

    \u09b8\u09cd\u09aa\u09b7\u09cd\u099f\u09a4, \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0<\/span> \\delta A<\/span><\/span>\u00a0\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0<\/span>\u00a0 y\u03b4x<\/span> <\/span>\u098f\u09b0 \u0985\u09aa\u09c7\u0995\u09cd\u09b7\u09be \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09b0 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u00a0 (y+\u03b4y)\u03b4x<\/span><\/span>\u00a0<\/span>\u00a0<\/span>\u0985\u09aa\u09c7\u0995\u09cd\u09b7\u09be <\/span>\u00a0<\/span>\u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09b0\u0964<\/span><\/p>\n

    \u0985\u09b0\u09cd\u09a5\u09be\u09ce, y \\delta x<\\delta A<(y+\\delta y) \\delta x<\/span><\/span><\/p>\n\\Rightarrow y<\\frac{\\delta A}{\\delta x}<y+\\delta y \\quad \\therefore \\lim _{\\delta y \\rightarrow 0} y<\\lim _{\\delta x \\rightarrow 0} \\frac{\\delta A}{\\delta x}<\\lim _{\\delta y \\rightarrow 0}(y+\\delta y) \\Rightarrow y<\\frac{d A}{d x}<y<\/span>\n

     <\/p>\n

    \\therefore \\frac{d A}{d x}=y \\Rightarrow d A=y d x<\/span><\/span><\/p>\n

    \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, A=\\int y d x=\\int F(x)+c<\/span> <\/span>(\u09a7\u09b0\u09bf)<\/span><\/p>\n

    x = a<\/span> \u09b9\u09b2\u09c7, <\/span>A = 0<\/span> \u2234 0=F(a)+c \\Rightarrow c=-F(a)<\/span><\/span><\/p>\n

    x = b<\/span> \u09b9\u09b2\u09c7, <\/span>A = ABCD<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0985\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2\u0964<\/span><\/p>\n

    \u2234ABCD<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0986\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 =F(b)+c=F(b)-F(a)=\\int_{a}^{b} y d x<\/span><\/span><\/p>\n

    \u0985\u09a4\u098f\u09ac, \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c<\/span>\u00a0<\/span> \\int_{a}^{b} y d x=\\int_{a}^{b} f(x) d x, y=f(x)<\/span>\u00a0\u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be, <\/span>x<\/span>-\u0985\u0995\u09cd\u09b7 \u098f\u09ac\u0982 <\/span>x = a\u00a0<\/span> \u0993 <\/span>x = b\u00a0<\/span> \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u0995\u09cb\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0986\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

    \u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4<\/b>:<\/span> \u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7 \u09a6\u09c7\u0996\u09be\u09a8\u09cb \u09af\u09be\u09af\u09bc,<\/span> \\int_{c}^{d} x d y<\/span><\/span>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c\u099f\u09bf \u098f\u0995\u099f\u09bf \u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be, <\/span>y<\/span>-\u0985\u0995\u09cd\u09b7 \u098f\u09ac\u0982 \u09a6\u09c1\u0987\u099f\u09bf \u09ad\u09c2\u099c <\/span>y = c<\/span>\u00a0<\/span>\u0993 <\/span>y = d<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0986\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

    <\/h1>\n

    \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be \u098f\u09ac\u0982 \u09a6\u09c1\u0987\u099f\u09bf \u0995\u09cb\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0986\u09ac\u09a6\u09cd\u09a7 \u09b8\u09ae\u09a4\u09b2 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 <\/span>=<\/span><\/p>\n

    y_{1}=f_{1}(x) 3 y_{2}=f_{2}(x)<\/span> \u00a0<\/span>\u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be \u098f\u09ac\u0982 <\/span>x = a<\/span> \u0993 <\/span>x = b<\/span> \u0995\u09cb\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0986\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2<\/span><\/p>\n

    =\\int_{a}^{b}\\left(y_{1}-y_{2}\\right) d x=\\int_{a}^{b}\\left\\{f_{1}(x)-f_{2}(x)\\right\\} d x<\/span>\u00a0<\/span><\/p>\n

    \u099a\u09bf\u09a4\u09cd\u09b0 \u09a5\u09c7\u0995\u09c7 \u098f\u099f\u09be \u09b8\u09cd\u09aa\u09b7\u09cd\u099f \u09af\u09c7, <\/span>ABCD<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2<\/span>\u00a0<\/span>= DMNC<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2<\/span>\u00a0<\/span>– AMNB<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 =\\int_{a}^{b} f_{1}(x) d x-\\int_{a}^{b} f_{2}(x) d x<\/span><\/span>,<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span>DC<\/span> \u0993 <\/span>AB<\/span> \u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be \u09a6\u09c1\u099f\u09bf\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7<\/span>\u00a0<\/span><\/p>\ny_{1}=f\\left(x_{1}\\right) \\text { \\& } y_{2}=f\\left(x_{2}\\right) \\text { \u098f\u09ac\u0982 } O M=a \\text { \\& } \\text { ON }=b<\/span>\n

     <\/p>\n

    \u2234<\/span> \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09af\u09bc \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 =\\int_{a}^{b}\\left(y_{1}-y_{2}\\right) d x=\\int_{a}^{b}\\left\\{f_{1}(x)-f_{2}(x)\\right\\} d x<\/span><\/span>\u00a0<\/span><\/p>\n

     <\/p>\n

    \u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e7: <\/b>x^{2}+y^{2}=16<\/span>\u00a0\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964<\/b> (Area of circle)<\/b><\/p>\n

    \u09b8\u09ae\u09be\u09a7\u09be\u09a8:<\/b><\/p>\n

    <\/h1>\n

    x^{2}+y^{2}=4^{2}<\/span> \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 <\/span>4<\/span>\u0964<\/span><\/p>\nx^{2}+y^{2}=16 \\Rightarrow y^{2}=16-x^{2} \\Rightarrow y=\\pm \\sqrt{16-x^{2}}<\/span>\n

    \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0 <\/span>OAB<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 =y=\\sqrt{16-x^{2}}<\/span><\/span>\u00a0\u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be, <\/span>x<\/span>-\u0985\u0995\u09cd\u09b7 \u098f\u09ac\u0982<\/span>\u00a0<\/span>x = 0<\/span> \u0993 <\/span>x = 4<\/span> \u0995\u09cb\u099f\u09bf \u09a6\u09c1\u0987\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c0\u09ae\u09be\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 = \\int_{0}^{4} y d x<\/span><\/span><\/p>\n

    =\\int_{0}^{4} \\sqrt{16-x^{2}} d x<\/span><\/span><\/p>\n

    =\\int_{0}^{4} \\sqrt{4^{2}-x^{2}} d x<\/span><\/span><\/p>\n

    =\\left[\\frac{x \\sqrt{4^{2}-x^{2}}}{2}+\\frac{4^{2}}{2} \\sin ^{-1} \\frac{x}{4}\\right]_{0}^{4}<\/span><\/span><\/p>\n

    =\\left[\\frac{4 \\sqrt{4^{2}-4^{2}}}{2}+\\frac{4^{2}}{2} \\sin ^{-1} \\frac{4}{4}\\right]-\\left[\\frac{0 \\sqrt{4^{2}-0^{2}}}{2}+\\frac{4^{2}}{2} \\sin ^{-1} \\frac{0}{4}\\right]<\/span><\/span><\/p>\n

    =\\frac{16}{2} \\sin ^{-1} 1<\/span><\/span><\/p>\n

    =8 \\cdot \\frac{\\pi}{2}<\/span><\/span><\/p>\n

    = 4\u03c0<\/span><\/p>\n

    \u2234<\/span> \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 <\/span>= 4 \u00d7 <\/span> \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0 <\/span>OAB<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 <\/span>= 4 <\/span>\u00d7 4\u03c0 = 16\u03c0<\/span>\u00a0<\/span>\u09ac\u09b0\u09cd\u0997 \u098f\u0995\u0995\u0964<\/span> (Ans)<\/b><\/p>\n

     <\/p>\n

    \u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e8: \\frac{x^{2}}{3^{2}}+\\frac{y^{2}}{2^{2}}=1<\/span><\/span><\/b>\u00a0\u0989\u09aa\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964\u00a0 (Elliptical Area)<\/b><\/p>\n

    \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n

    <\/h1>\n

    \\frac{x^{2}}{3^{2}}+\\frac{y^{2}}{2^{2}}=1<\/span><\/b> \u0989\u09aa\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u098f\u09ac\u0982 \u09ac\u09c3\u09b9\u09ce \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af\u09c7\u09b0 \u0985\u09b0\u09cd\u09a7\u09c7\u0995<\/span>\u00a0<\/span>3<\/span> \u098f\u0995\u0995\u0964<\/span><\/p>\n

    \\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1 \\Rightarrow \\frac{y^{2}}{4}=1-\\frac{x^{2}}{9} \\Rightarrow y^{2}=\\frac{4}{9}\\left(9-x^{2}\\right)<\/span> <\/span><\/p>\n

    \\Rightarrow y=\\pm \\frac{2}{3} \\sqrt{9-x^{2}}<\/span><\/span><\/p>\n

    \u2234<\/span> \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0 <\/span>OAB<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 =y=\\frac{2}{3} \\sqrt{9-x^{2}}<\/span><\/span>\u00a0\u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be, <\/span>x<\/span>-\u0985\u0995\u09cd\u09b7 \u098f\u09ac\u0982<\/span>\u00a0<\/span>x = 0<\/span> \u0993<\/span>\u00a0<\/span>x = 3<\/span> \u0995\u09cb\u099f\u09bf\u09a6\u09cd\u09ac\u09af\u09bc \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c0\u09ae\u09be\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 <\/span>= \\int_{0}^{3} y d x<\/span><\/span><\/p>\n

    =\\int_{0}^{3} \\frac{2}{3} \\sqrt{9-x^{2}} d x<\/span><\/span><\/p>\n

    =\\frac{2}{3} \\int_{0}^{3} \\sqrt{3^{2}-x^{2}} d x<\/span><\/span><\/p>\n

    =\\frac{2}{3}\\left[\\frac{x \\sqrt{3^{2}-x^{2}}}{2}+\\frac{3^{2}}{2} \\sin ^{-1} \\frac{x}{3}\\right]_{0}^{3}<\/span><\/span><\/p>\n

    =\\frac{2}{3}\\left[\\frac{3 \\sqrt{3^{2}-3^{2}}}{2}+\\frac{3^{2}}{2} \\sin ^{-1} \\frac{3}{3}\\right]-\\frac{2}{3}\\left[\\frac{0 \\sqrt{3^{2}-0^{2}}}{2}+\\frac{3^{2}}{2} \\sin ^{-1} \\frac{0}{3}\\right]<\/span><\/span><\/p>\n

    =\\frac{2}{3}\\left\\{\\frac{9}{2} \\sin ^{-1} 1\\right\\}=\\frac{2}{3} \\times \\frac{9}{2} \\times \\frac{\\pi}{2}=\\frac{3 \\pi}{2}<\/span><\/span><\/p>\n

    \u2234<\/span> \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u0989\u09aa\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 <\/span>= 4 \u00d7<\/span> \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0 <\/span>OAB<\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2<\/span><\/p>\n

    = 4 \\times \\frac{3 \\pi}{2}<\/span><\/span>\u09ac\u09b0\u09cd\u0997 \u098f\u0995\u0995<\/span><\/p>\n

    = 6 \\pi<\/span><\/span>\u00a0\u09ac\u09b0\u09cd\u0997 \u098f\u0995\u0995<\/span><\/p>\n

     <\/p>\n

    \u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e9: \u09a6\u09c7\u0996\u09be\u0993 \u09af\u09c7, y^{2}=4 a x<\/span> <\/span><\/b>\u098f\u09ac\u0982 x^{2}=4 a y<\/span> <\/span><\/b>\u09aa\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4 \u09a6\u09c1\u0987\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c0\u09ae\u09be\u09ac\u09a6\u09cd\u09a7 \u09b8\u09ae\u09a4\u09b2 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 <\/b>\\frac{16}{3} a^{2}<\/span><\/p>\n

    \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n

    <\/h1>\n

    x^{2}=4 a y \\Rightarrow y=\\frac{x^{2}}{4 a}<\/span> \u09b9\u09a4\u09c7, <\/span>y<\/span> \u098f\u09b0 \u09ae\u09be\u09a8 y^{2}=4 a x<\/span><\/span>\u00a0\u098f\u0987 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 \u09ac\u09b8\u09bf\u09df\u09c7 \u09aa\u09be\u0987,\u00a0<\/span><\/p>\n

    \\left(\\frac{x^{2}}{4 a}\\right)^{2}=4 a x<\/span><\/span><\/p>\n

    \\Rightarrow x^{4}=64 a^{3} x<\/span><\/span><\/p>\n

    \\Rightarrow x\\left(x^{3}-64 a^{3}\\right)=0<\/span><\/span><\/p>\n

    \\Rightarrow x=0,4 a<\/span><\/span><\/p>\n

    \u098f\u0996\u09be\u09a8\u09c7, <\/span>x<\/span> \u098f\u09b0 \u09b8\u09c0\u09ae\u09be <\/span>0<\/span> \u09a5\u09c7\u0995\u09c7 <\/span>4a<\/span> \u098f\u09ac\u0982 y_{1}=2 \\sqrt{a} \\sqrt{x}, y_{2}=\\frac{1}{4 a} x^{2}<\/span><\/span><\/p>\n

    \u2234<\/span> \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 = \\int_{0}^{1}\\left(y_{1}-y_{2}\\right) d x<\/span><\/span><\/p>\n

    = \\int_{0}^{4 a}\\left(2 \\sqrt{a} \\sqrt{x}-\\frac{1}{4 a} x^{2}\\right) d x<\/span><\/span><\/p>\n

    =\\left[2 \\sqrt{a} \\frac{2}{3} x^{3 \/ 2}-\\frac{1}{4 a} \\cdot \\frac{x^{3}}{3}\\right]_{0}^{4 a}<\/span><\/span><\/p>\n

    =2 \\sqrt{a} \\frac{2}{3}(4 a)^{3 \/ 2}-\\frac{1}{12 a}(4 a)^{3}<\/span><\/span><\/p>\n

    =\\frac{4 \\sqrt{a}}{3} \\times 8 a \\sqrt{a}-\\frac{1}{12 a} \\cdot 64 a^{3}<\/span><\/span><\/p>\n

    =\\frac{32}{3} a^{2}-\\frac{16}{3} a^{2}<\/span><\/span><\/p>\n

    =\\frac{32-16}{3} a^{2}<\/span><\/span><\/p>\n

    =\\frac{16}{3} a^{2}<\/span> \u09ac\u09b0\u09cd\u0997 \u098f\u0995\u0995<\/span> (\u09a6\u09c7\u0996\u09be\u09a8\u09cb \u09b9\u09b2\u09cb)<\/b><\/p>\n

     <\/p>\n","protected":false},"excerpt":{"rendered":"

    \u09ae\u09c2\u09b2\u09a6 \u09ac\u09c0\u099c\u0997\u09a3\u09bf\u09a4\u09c0\u09af\u09bc \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6\u09c7\u09b0 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 (Integration of Rational Algebraic Fractions): \u0995\u09cb\u09a8\u09cb \u09ae\u09c2\u09b2\u09a6 \u09ac\u09c0\u099c\u0997\u09a3\u09bf\u09a4\u09c0\u09af\u09bc \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6\u09c7\u09b0 \u09af\u09cb\u0997\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09be\u09b0 \u099c\u09a8\u09cd\u09af \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6\u099f\u09bf\u0995\u09c7 \u0986\u0982\u09b6\u09bf\u0995 \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6\u09c7 \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3 \u0995\u09b0\u09c7 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995 \u0985\u0982\u09b6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09af\u09cb\u099c\u09bf\u09a4 \u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09af\u09bc\u0964 \u09a8\u09bf\u09af\u09bc\u09ae (Rule):   \u0985\u09ad\u09bf\u099c\u09cd\u099e\u09a4\u09be\u09b2\u09ac\u09cd\u09a7 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf (Thumb Rule Method): \u09af\u09a6\u09bf \u0995\u09cb\u09a8\u09cb<\/p>\n

    Read More<\/a><\/p>\n","protected":false},"author":56,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[4265,3037,50,3026],"tags":[2379,2381,2380,2378,2382,2383],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3364"}],"collection":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/users\/56"}],"replies":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/comments?post=3364"}],"version-history":[{"count":35,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3364\/revisions"}],"predecessor-version":[{"id":16135,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3364\/revisions\/16135"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=3364"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=3364"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=3364"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}