{"id":3363,"date":"2024-01-30T12:14:00","date_gmt":"2024-01-30T06:14:00","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3363"},"modified":"2024-11-05T17:15:14","modified_gmt":"2024-11-05T11:15:14","slug":"trigonometric-functions","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/trigonometric-functions\/","title":{"rendered":"\u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09af\u09cb\u099c\u09bf\u09a4 \u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df, \u0985\u0982\u09b6\u09be\u09df\u09a8\u09c7\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3"},"content":{"rendered":"

\u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09af\u09cb\u099c\u09bf\u09a4 \u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u09b8\u09c2\u09a4\u09cd\u09b0 \u0985\u09a8\u09c1\u09af\u09be\u09df\u09c0<\/b><\/span><\/h2>\n

Determining the Combined Results of Trigonometric Functions<\/strong><\/span><\/h2>\n

\u09a7\u09b0\u09a8-\u09ef (Type \u2013 9):<\/b><\/p>\n

(a)<\/b> \\int \\sin ^{n} x d x, \\int \\cos ^{n} x d x<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7, <\/span>n\u2208N<\/span><\/p>\n

\u09aa\u09cd\u09b0\u09a4\u09bf\u09b8\u09cd\u09a5\u09be\u09aa\u09a8<\/b>: <\/span>(i)<\/b> n<\/span> \u09ac\u09bf\u099c\u09cb\u09dc \u09b9\u09b2\u09c7, \\int \\sin ^{n} x d x<\/span><\/span> \u098f\u09b0 \u099c\u09a8\u09cd\u09af \\cos x=z<\/span><\/span> \u098f\u09ac\u0982 \\int \\cos ^{n} x d x<\/span> <\/span>\u098f\u09b0 \u099c\u09a8\u09cd\u09af <\/span>sinx = z<\/span> \u09a7\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

(ii)<\/b> n<\/span> \u099c\u09cb\u09dc \u09b9\u09b2\u09c7, \\sin ^{2} x=\\frac{1}{2}(1-\\cos 2 x)<\/span> <\/span>\u098f\u09ac\u0982 \\cos ^{2} x=\\frac{1}{2}(1+\\cos 2 x)<\/span><\/span>\u00a0\u09b8\u09c2\u09a4\u09cd\u09b0 \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e7\u09e8: \\int \\sin ^{5} x d x<\/span><\/b> \u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

Example \u2013 12: Determine \\int \\sin ^{5} x d x<\/span><\/b><\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n\\int \\sin ^{5} x d x=\\int \\sin ^{4} x \\sin x d x=\\int\\left(1-\\cos ^{2} x\\right)^{2} \\sin x d x<\/span>\n

 <\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>cosx=z<\/span> \\therefore-\\sin x d x=d z \\Rightarrow \\sin x d x=-d z<\/span><\/p>\n\\therefore \\int \\sin ^{5} x d x=\\int\\left(1-z^{2}\\right)^{2}(-d z)<\/span>\n

 <\/p>\n

=-\\int\\left(1-2 z^{2}+z^{4}\\right) d z<\/span><\/span><\/p>\n

=-\\left(z+2 \\cdot \\frac{z^{3}}{3}+\\frac{z^{5}}{5}\\right)+c<\/span><\/span><\/p>\n

=-\\cos x+\\frac{2}{3} \\cos ^{3} x-\\frac{1}{5} \\cos ^{5} x+c<\/span><\/span>\u00a0(Ans)<\/b><\/p>\n

 <\/p>\n

(b)<\/b> \\int \\sin ^{m} x \\cos ^{n} x d x<\/span><\/p>\n

\u09aa\u09cd\u09b0\u09a4\u09bf\u09b8\u09cd\u09a5\u09be\u09aa\u09a8<\/b>: <\/span>(i)<\/b> m,\u00a0n\u2208N<\/span> \u098f\u09ac\u0982 \u0989\u09ad\u09df\u09c7 \u09ac\u09bf\u099c\u09cb\u09dc \u09b9\u09b2\u09c7, <\/span>m>n<\/span> \u098f\u09b0 \u099c\u09a8\u09cd\u09af <\/span>sinx= z<\/span> \u098f\u09ac\u0982 <\/span>n>m<\/span> \u098f\u09b0 \u099c\u09a8\u09cd\u09af <\/span>cosx= z<\/span> \u09a7\u09b0\u09a4\u09c7 \u09b9\u09df\u0964 <\/span>m<\/span> \u09ac\u09bf\u099c\u09cb\u09dc \u0993 <\/span>n<\/span> \u099c\u09cb\u09dc \u09b9\u09b2\u09c7, <\/span>cosx= z<\/span>; <\/span>n<\/span> \u09ac\u09bf\u099c\u09cb\u09dc \u0993 <\/span>m<\/span> \u099c\u09cb\u09dc \u09b9\u09b2\u09c7, <\/span>sinx= z<\/span> \u09a7\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

(ii) <\/b>m,\u00a0n\u2208N<\/span> \u098f\u09ac\u0982 \u0989\u09ad\u09df\u09c7 \u099c\u09cb\u09dc \u09b9\u09b2\u09c7, \\sin ^{2} x=\\frac{1}{2}(1-\\cos 2 x)<\/span><\/span> \u098f\u09ac\u0982 \\cos ^{2} x=\\frac{1}{2}(1+\\cos 2 x)<\/span><\/span>\u00a0\u09b8\u09c2\u09a4\u09cd\u09b0 \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

(iii) <\/b>m<\/span> \u0993 <\/span>n<\/span> \u0989\u09ad\u09df\u09c7 \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6 \u09b9\u09b2\u09c7, \u098f\u09ac\u0982 <\/span>-(m+n) \u2208 N<\/span> \u0993 \u099c\u09cb\u09dc \u09b9\u09b2\u09c7, \u09b2\u09ac \u0993 \u09b9\u09b0 \u0989\u09ad\u09df\u0995\u09c7 \\sec ^{-(m+n)} x<\/span> <\/span>\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e7\u09e9:\\int \\frac{\\sqrt{\\tan x}}{\\sin x \\cos x} d x<\/span>\u00a0 <\/b>\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

Example 13 – \\int \\frac{\\sqrt{\\tan x}}{\\sin x \\cos x} d x<\/span><\/b><\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n\\int \\frac{\\sqrt{\\tan x}}{\\sin x \\cos x} d x=\\int \\frac{\\sqrt{\\sin x}}{\\sqrt{\\cos x} \\sin x \\cos x} d x<\/span>\n

 <\/p>\n=\\int \\sin ^{-1 \/ 2} x \\cos ^{-3 \/ 2} x d x<\/span>\n

 <\/p>\n=\\int \\frac{\\sec ^{2} x}{\\sec ^{2} x}\\left(\\sin ^{-1 \/ 2} x \\cos ^{-3 \/ 2} x\\right) d x \\quad\\left[\\because-\\left(-\\frac{1}{2}-\\frac{3}{2}\\right)=2 \\in \\mathbb{N}\\right]<\/span>\n

 <\/p>\n=\\int \\frac{\\sec ^{2} x}{\\sec ^{2} x \\sin ^{1 \/ 2} x \\cos ^{3 \/ 2} x} d x<\/span>\n

 <\/p>\n=\\int \\frac{\\sec ^{2} x}{\\frac{\\sin ^{1 \/ 2} x}{\\cos ^{1 \/ 2} x} \\frac{\\cos ^{3 \/ 2} x}{\\cos ^{3 \/ 2} x}} d x<\/span>\n

 <\/p>\n=\\int \\frac{d(\\tan x)}{\\sqrt{\\tan x}}<\/span>\n

 <\/p>\n

=2 \\sqrt{\\tan x}+c<\/span>\u00a0 \u00a0 (Ans)<\/b><\/p>\n

\u09ac\u09bf\u0995\u09b2\u09cd\u09aa \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Alternate solution):<\/b><\/p>\n\\int \\frac{\\sqrt{\\tan x}}{\\sin x \\cos x} d x=\\int \\frac{\\sqrt{\\tan x}}{\\frac{\\sin x}{\\operatorname{cosx}} \\cos ^{2} x} d x<\/span>\n

 <\/p>\n=\\int \\frac{\\sqrt{\\tan x}}{\\tan x \\cos ^{2} x} d x<\/span>\n

 <\/p>\n=\\int \\frac{\\sec ^{2} x}{\\sqrt{\\tan x}} d x<\/span>\n

 <\/p>\n=\\int \\frac{d(\\tan x)}{\\sqrt{\\tan x}}<\/span>\n

 <\/p>\n

=2 \\sqrt{\\tan x}+c<\/span>\u00a0 (Ans)<\/b><\/p>\n

 <\/p>\n

\u09a7\u09b0\u09a8-\u09e7\u09e6 (Type – 10):<\/b><\/p>\n

\\int \\tan ^{n} x d x, \\int \\cot ^{n} x d x<\/span>,<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7, <\/span>n \u2208 N<\/span><\/p>\n

\u09a8\u09bf\u09df\u09ae (Rule)<\/b>: \\int \\tan ^{n} x d x=\\int \\tan ^{n-2} x \\tan ^{2} x d x<\/span><\/span><\/p>\n

=\\int \\tan ^{n-2} x\\left(\\sec ^{2} x-1\\right) d x<\/span><\/span><\/p>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e7\u09ea: <\/b>\\int \\tan ^{3} x d x<\/span>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

Example \u2013 14: Determine \\int \\tan ^{3} x d x<\/span><\/b><\/p>\n

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

\\int \\tan ^{3} x d x=\\int \\tan x \\tan ^{2} x d x<\/span>\u00a0<\/span><\/p>\n

=\\int \\tan x\\left(\\sec ^{2} x-1\\right) d x<\/span><\/span><\/p>\n

=\\int \\tan x \\sec ^{2} x d x-\\int \\tan x d x<\/span><\/span><\/p>\n

=\\int \\tan x d(\\tan x)-\\int \\tan x d x<\/span><\/span><\/p>\n

=\\frac{(\\tan x)^{2}}{2}-(-\\ln |\\cos x|)+c<\/span>\u00a0\u00a0<\/span><\/p>\n

=\\frac{\\tan ^{2} x}{2}+\\ln |\\cos x|+c<\/span> <\/span>\u00a0(Ans)<\/b><\/p>\n

 <\/p>\n

\u09a7\u09b0\u09a8-\u09e7\u09e7 (Type \u2013 11):<\/b><\/p>\n

(a)\\int \\frac{d x}{a+b \\cos x+c \\sin x}, \\int \\frac{d x}{a+b \\cos x}, \\int \\frac{d x}{a+b \\sin x}<\/span><\/b><\/p>\n

\u09a8\u09bf\u09df\u09ae (Rule)<\/b>: \\sin x=\\frac{2 \\tan \\frac{x}{2}}{1+\\tan ^{2} \\frac{x}{2}}<\/span><\/span> \u098f\u09ac\u0982 \\cos x=\\frac{1-\\tan ^{2} \\frac{x}{2}}{1+\\tan ^{2} \\frac{x}{2}}<\/span><\/span>\u00a0\u09b8\u09c2\u09a4\u09cd\u09b0 \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

(b)<\/b> \\int \\frac{d x}{a \\sin x+b \\cos x}, \\int \\frac{d x}{(a \\sin x+b \\cos x)^{n}}<\/span><\/span><\/p>\n

\u09a8\u09bf\u09df\u09ae (Rule)<\/b>: a=r \\cos \\theta<\/span><\/span> \u098f\u09ac\u0982 b=r \\sin \\theta<\/span><\/span>\u00a0\u09a7\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

(c)<\/b> \\int \\frac{c \\cos x d x}{a \\sin x+b \\cos x}, \\int \\frac{c \\sin x d x}{a \\sin x+b \\cos x}, \\int \\frac{c \\sin x+d \\cos x}{a \\sin x+b \\cos x} d x<\/span><\/span><\/p>\n

\u09a8\u09bf\u09df\u09ae (Rule)<\/b>: \u09b2\u09ac\u0995\u09c7 <\/span>l<\/span> (\u09b9\u09b0) +m \\frac{d}{d x}<\/span><\/span>\u00a0(\u09b9\u09b0) \u098f\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

(d)<\/b> \\int \\frac{p \\sin x+q \\cos x+r}{a \\sin x+b \\cos x+c} d x<\/span><\/span><\/p>\n

\u09a8\u09bf\u09df\u09ae (Rule)<\/b>: \u09b2\u09ac\u0995\u09c7 <\/span>l<\/span> (\u09b9\u09b0) +m \\frac{d}{d x}<\/span><\/span>\u00a0(\u09b9\u09b0) +<\/span>n<\/span> \u098f\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

(e)<\/b> \\int \\frac{d x}{\\sin ^{m} x \\cos ^{n} x}<\/span> <\/span><\/p>\n

\u09a8\u09bf\u09df\u09ae (Rule)<\/b>: d x=\\left(\\sin ^{2} x+\\cos ^{2} x\\right) d x<\/span><\/span>\u00a0\u09b2\u09bf\u0996\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e7\u09eb: <\/b>\\int \\frac{1}{1+\\tan x} d x<\/span>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

Example \u2013 15: Determine \\int \\frac{1}{1+\\tan x} d x<\/span><\/b><\/p>\n

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

\\int \\frac{1}{1+\\tan x} d x=\\int \\frac{1}{1+\\frac{\\sin x}{\\cos x}} d x=\\int \\frac{\\cos x}{\\sin x+\\cos x} d x<\/span> <\/span><\/p>\n

=\\int \\frac{1}{2} \\cdot \\frac{(\\sin x+\\cos x)+(\\cos x-\\sin x)}{\\sin x+\\cos x} d x<\/span><\/span><\/p>\n

=\\frac{1}{2}\\left[\\int \\frac{\\sin x+\\cos x}{\\sin x+\\cos x} d x+\\int \\frac{\\cos x-\\sin x}{\\sin x+\\cos x} d x\\right]<\/span><\/span><\/p>\n

=\\frac{1}{2}\\left[\\int d x+\\int \\frac{d(\\sin x+\\cos x)}{\\sin x+\\cos x}\\right]<\/span><\/span>\u00a0<\/span><\/p>\n

=\\frac{1}{2}[x+\\ln |\\sin x+\\cos x|]+c<\/span>\u00a0 <\/span>(Ans)<\/b><\/p>\n

\u09ac\u09bf\u0995\u09b2\u09cd\u09aa \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Alternate Solution):<\/b><\/p>\n\\int \\frac{1}{1+\\tan x} d x=\\frac{1}{2} \\int \\frac{(1+\\tan x)+(1-\\tan x)}{1+\\tan x} d x<\/span>\n

 <\/p>\n

=\\frac{1}{2} \\int\\left[1+\\frac{1-\\tan x}{1+\\tan x}\\right] d x<\/span><\/span><\/p>\n

=\\frac{1}{2} \\int\\left[1+\\frac{1-\\frac{\\sin x}{\\operatorname{cosx}}}{1+\\frac{\\sin x}{\\operatorname{cosx}}}\\right] d x<\/span><\/span><\/p>\n

=\\frac{1}{2} \\int\\left[1+\\frac{\\cos x-\\sin x}{\\cos x+\\sin x}\\right] d x<\/span><\/span><\/p>\n

=\\frac{1}{2}\\left[\\int d x+\\int \\frac{\\cos x-\\sin x}{\\sin x+\\cos x} d x\\right]<\/span><\/span><\/p>\n

=\\frac{1}{2}\\left[\\int d x+\\int \\frac{d(\\sin x+\\cos x)}{\\sin x+\\cos x}\\right]<\/span><\/span><\/p>\n

=\\frac{1}{2}[x+\\ln |\\sin x+\\cos x|]+c<\/span><\/span>\u00a0(Ans)<\/b><\/p>\n

 <\/p>\n

\u09a7\u09b0\u09a8-\u09e7\u09e8 (Type – 12):<\/b><\/p>\n

(a)<\/b> \\int \\frac{c \\sin 2 x d x}{a \\cos ^{2} x+b \\sin ^{2} x}<\/span><\/p>\n

\u09a8\u09bf\u09df\u09ae (Rule)<\/b>: z=a \\cos ^{2} x+b \\sin ^{2} x<\/span><\/span>\u00a0\u09a7\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

(b) \\int \\frac{d x}{a \\cos ^{2} x+b \\sin ^{2} x}, \\int \\frac{d x}{a+b \\sin ^{2} x}, \\int \\frac{d x}{a+b \\cos ^{2} x}<\/span><\/b><\/p>\n

\u09a8\u09bf\u09df\u09ae (Rule)<\/b>: \u09b2\u09ac \u0993 \u09b9\u09b0 \u0989\u09ad\u09df\u0995\u09c7 \\cos ^{2} x<\/span><\/span>\u00a0\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e7\u09ec: \\int \\frac{d \\theta}{1+3 \\cos ^{2} \\theta}<\/span> <\/b>\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

Example \u2013 16: Determine \\int \\frac{d \\theta}{1+3 \\cos ^{2} \\theta}<\/span><\/b><\/p>\n

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

 <\/p>\n

=\\int \\frac{\\sec ^{2} \\theta d \\theta}{\\sec ^{2} \\theta+3}<\/span><\/span><\/p>\n

=\\int \\frac{\\sec ^{2} \\theta d \\theta}{1+\\tan ^{2} \\theta+3}<\/span><\/span>\u00a0<\/span><\/p>\n

=\\int \\frac{d(\\tan \\theta)}{2^{2}+(\\tan \\theta)^{2}}<\/span><\/span><\/p>\n

=\\frac{1}{2} \\tan ^{-1}\\left(\\frac{\\tan \\theta}{2}\\right)+c<\/span><\/span>\u00a0(Ans)<\/b><\/p>\n

 <\/p>\n

\u0985\u0982\u09b6\u09be\u09df\u09a8\u09c7\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 (Integration by parts):<\/b><\/p>\n

u<\/span> \u0993 <\/span>v<\/span> \u0989\u09ad\u09af\u09bc\u09c7 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3\u09af\u09cb\u0997\u09cd\u09af <\/span>x<\/span> \u098f\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u09ab\u09be\u0982\u09b6\u09a8<\/a><\/span> \u09b9\u09b2\u09c7, \u098f \u09ac\u09bf\u09b6\u09c7\u09b7 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 <\/span>\\int u v d x<\/span><\/span> \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964<\/span>\u00a0<\/span>uw<\/span> \u0995\u09c7 <\/span>x<\/span> \u098f\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, \\frac{d}{d x}(u w)=u \\frac{d w}{d x}+w \\frac{d u}{d x}<\/span><\/span><\/p>\n

\u0987\u09b9\u09be\u0995\u09c7 <\/span>x<\/span> \u098f\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

u w=\\int u \\frac{d w}{d x} d x+\\int w \\frac{d u}{d x} d x<\/span><\/span><\/p>\n

\\Rightarrow \\int u \\frac{d w}{d x} d x=u w-\\int w \\frac{d u}{d x} d x \\ldots \\ldots \\text { (i) }<\/span><\/span><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \\frac{d w}{d x}=v \\quad \\therefore w=\\int v d x<\/span><\/span><\/p>\n

(i)<\/span> \u09a5\u09c7\u0995\u09c7 \u09aa\u09be\u0987,<\/span> \\int u v d x=u \\int v d x-\\int\\left\\{\\frac{d u}{d x} \\int v d x\\right\\} d x<\/span><\/span><\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09a6\u09c1\u0987\u099f\u09bf \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2\u09c7\u09b0 \u09af\u09cb\u0997\u099c <\/span>=<\/span> \u09aa\u09cd\u09b0\u09a5\u09ae \u09ab\u09be\u0982\u09b6\u09a8 x <\/span>\u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09af\u09cb\u0997\u099c <\/span>-(<\/span>\u09aa\u09cd\u09b0\u09a5\u09ae \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u0995 \u09b8\u09b9\u0997 <\/span> \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09af\u09cb\u0997\u099c) \u098f\u09b0 \u09af\u09cb\u0997\u099c\u0964<\/span><\/p>\n

 <\/p>\n

uv<\/span> \u09a7\u09b0\u09be\u09b0 \u0995\u09cc\u09b6\u09b2 (Tricks to find <\/b>uv<\/span>):<\/b><\/p>\n

(i) <\/b>\u2018LIATE\u2019 \u09b6\u09ac\u09cd\u09a6\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09b9\u099c\u09c7\u0987 \u0995\u09cb\u09a8\u099f\u09bf <\/span>u<\/span> \u0993 \u0995\u09cb\u09a8\u099f\u09bf <\/span>v<\/span> \u09b9\u09ac\u09c7 \u09a4\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964 L(Logarithmic), I(Inverse), A(Algebraic), T(Trigonometric), E(Exponential)<\/span>\u00a0<\/span>\u0995\u09cd\u09b0\u09ae\u09be\u09a8\u09c1\u09af\u09be\u09af\u09bc\u09c0 \u09af\u09c7\u099f\u09bf \u0986\u0997\u09c7 \u09b8\u09c7\u099f\u09bf\u0995\u09c7 <\/span>u<\/span> \u098f\u09ac\u0982 \u09af\u09c7\u099f\u09bf \u09aa\u09b0\u09c7 \u09b8\u09c7\u099f\u09bf\u0995\u09c7 <\/span>v<\/span> \u09a7\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n

(ii) <\/b>\u09b8\u09be\u09a7\u09be\u09b0\u09a3\u09a4 \u09ac\u09c0\u099c\u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u0995\u09c7\\left(x^{n}\\right)<\/span><\/span>\u00a0\u09aa\u09cd\u09b0\u09a5\u09ae \u09ab\u09be\u0982\u09b6\u09a8 \u09a7\u09b0\u09a4\u09c7 \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

(iii) \\int \\ln x d x, \\int \\sin ^{-1} x d x<\/span><\/span><\/b>\u00a0\u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0986\u0995\u09be\u09b0\u09c7\u09b0 \u09af\u09cb\u0997\u099c\u09c7\u09b0 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u099c\u09a8\u09cd\u09af<\/span>\u00a0<\/span>1<\/span> \u0995\u09c7 \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc \u09ab\u09be\u0982\u09b6\u09a8 \u09b9\u09bf\u09b8\u09be\u09ac\u09c7 \u0997\u09a3\u09cd\u09af \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

 <\/p>\n

\u098f\u0995\u099f\u09bf \u09ac\u09bf\u09b6\u09c7\u09b7 \u09b8\u09c2\u09a4\u09cd\u09b0 (Special Formula): \\int e^{a x}\\left\\{a f(x)+f^{\\prime}(x)\\right\\} d x=e^{a x} f(x)+c \\text { i. e. }<\/span><\/span><\/b><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \\int e^{a x}[a f(x)+D\\{f(x)\\}] d x=e^{a x} f(x)+c<\/span><\/span><\/p>\n

\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 (Proof)<\/b>:<\/span> \\frac{d}{d x}\\left\\{e^{a x} f(x)\\right\\}=f(x) \\frac{d}{d x}\\left(e^{a x}\\right)+e^{a x} \\frac{d}{d x} f(x)=f(x) \\cdot a e^{a x}+e^{a x} D\\{f(x)\\} \\quad\\left[\\because D=\\frac{d}{d x}\\right]<\/span><\/span><\/p>\n

\\therefore \\int e^{a x}[a f(x)+D\\{f(x)\\}] d x=e^{a x} f(x)+c . \\text { i. e., }<\/span> <\/span><\/p>\n

\\int e^{a x}\\left\\{a f(x)+f^{\\prime}(x)\\right\\} d x=e^{a x} f(x)+c<\/span><\/span><\/p>\n

a=1 \\text { \u09b9\u09b2\u09c7, } \\int e^{x}\\left\\{f(x)+f^{\\prime}(x)\\right\\} d x=e^{x} f(x)+c<\/span> <\/span><\/p>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e7: \\int x^{3} \\sin 2 x d x<\/span><\/span><\/b>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

Example \u2013 1: Determine \\int x^{3} \\sin 2 x d x<\/span><\/span><\/b><\/p>\n

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

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

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

=-\\frac{1}{2} x^{3} \\cos 2 x+\\frac{3}{2}\\left[x^{2} \\cdot \\frac{\\sin 2 x}{2}-\\int 2 x\\left(\\frac{\\sin 2 x}{2}\\right) d x\\right]<\/span><\/span><\/b><\/p>\n=-\\frac{1}{2} x^{3} \\cos 2 x+\\frac{3}{4} x^{2} \\sin 2 x-\\frac{3}{2}\\left[x \\int \\sin 2 x d x-\\int\\left\\{\\frac{d}{d x}(x) \\int \\sin 2 x d x\\right\\} d x\\right] <\/span>\n

=-\\frac{1}{2} x^{3} \\cos 2 x+\\frac{3}{4} x^{2} \\sin 2 x-\\frac{3}{2}\\left[x\\left(\\frac{-\\cos 2 x}{2}\\right)-\\int\\left(\\frac{-\\cos 2 x}{2}\\right) d x\\right]<\/span><\/span><\/b><\/p>\n

=-\\frac{1}{2} x^{3} \\cos 2 x+\\frac{3}{4} x^{2} \\sin 2 x+\\frac{3}{4}\\left[x \\cos 2 x-\\frac{\\sin 2 x}{2}\\right]+c<\/span><\/span><\/b><\/p>\n

=-\\frac{1}{2} x^{3} \\cos 2 x+\\frac{3}{4} x^{2} \\sin 2 x+\\frac{3}{4} x \\cos 2 x-\\frac{3}{8} \\sin 2 x+c<\/span> <\/span>(Ans)<\/b><\/p>\n

\u09ac\u09bf\u0995\u09b2\u09cd\u09aa \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Alternate Solution):<\/b><\/p>\n

\\int x^{3} \\sin 2 x d x<\/span><\/span><\/b><\/p>\n=x^{3}\\left(\\frac{-\\cos 2 x}{2}\\right)-\\left(3 x^{2}\\right)\\left(-\\frac{1}{2} \\cdot \\frac{\\sin 2 x}{2}\\right)+(6 x)\\left(-\\frac{1}{4} \\cdot \\frac{-\\cos 2 x}{2}\\right)-6\\left(\\frac{1}{8} \\cdot \\frac{\\sin 2 x}{2}\\right)+c<\/span>\n

=-\\frac{1}{2} x^{3} \\cos 2 x+\\frac{3}{4} x^{2} \\sin 2 x+\\frac{3}{4} x \\cos 2 x-\\frac{3}{8} \\sin 2 x+c<\/span><\/span><\/b> \u00a0(Ans)<\/b><\/p>\n

\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af (Note): <\/b>\u09aa\u09cd\u09b0\u09a5\u09ae \u09aa\u09a6\u09c7\u09b0 \u09aa\u09b0\u09ac\u09b0\u09cd\u09a4\u09c0 \u09aa\u09a6 \u09a5\u09c7\u0995\u09c7 \u09ac\u09c0\u099c\u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0<\/a><\/span> \u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u0995\u09cd\u09b0\u09ae\u09bf\u0995 \u0985\u09a8\u09cd\u09a4\u09b0\u0995 \u09b8\u09b9\u0997 \u0993 \u09aa\u09cd\u09b0\u09a5\u09ae \u09aa\u09a6 \u09a5\u09c7\u0995\u09c7 \u09a4\u09cd\u09b0\u09bf\u0995\u09c7\u09a3\u09cb\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u0995\u09cd\u09b0\u09ae\u09bf\u0995 \u09af\u09cb\u0997\u099c \u0993 \u09aa\u09cd\u09b0\u09a5\u09ae \u09aa\u09a6 \u09a5\u09c7\u0995\u09c7 \u099a\u09bf\u09b9\u09cd\u09a8\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u0964 \u098f \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf \u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u0995\u09cd\u09b0\u09ae\u09bf\u0995 \u0985\u0982\u09b6\u09be\u09af\u09bc\u09a8 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u201cSuccessive integration by parts\u201d.<\/span><\/p>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e8: <\/b>\\int x \\cos ^{2} x d x<\/span>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

Example \u2013 2: Determine \\int x \\cos ^{2} x d x<\/span>\u00a0<\/b><\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n\\int x \\cos ^{2} x d x=\\int \\frac{1}{2} x(1+\\cos 2 x) d x<\/span>\n

 <\/p>\n

=\\frac{1}{2}\\left[\\int x d x+\\int x \\cos 2 x d x\\right]<\/span><\/span><\/p>\n

=\\frac{1}{2}\\left[\\frac{x^{2}}{2}+x \\int \\cos 2 x d x-\\int\\left\\{\\frac{d}{d x}(x) \\int \\cos 2 x d x\\right\\} d x\\right.]<\/span><\/span><\/p>\n

=\\frac{1}{2}\\left[\\frac{x^{2}}{2}+x \\frac{\\sin 2 x}{2}-\\int 1 \\cdot \\frac{\\sin 2 x}{2} d x\\right.]<\/span><\/span><\/p>\n

=\\frac{1}{2}\\left[\\frac{x^{2}}{2}+\\frac{x \\sin 2 x}{2}-\\frac{1}{2} \\cdot\\left(\\frac{-\\cos 2 x}{2}\\right)\\right]+c<\/span><\/span><\/p>\n

=\\frac{1}{4}\\left[x^{4}+x \\sin 2 x+\\frac{1}{2} \\cos 2 x\\right]+c<\/span><\/span>\u00a0 \u00a0(Ans)<\/b><\/p>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e9: \\int \\ln x d x, x>0<\/span><\/b>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

Example \u2013 3: Determine \\int \\ln x d x, x>0<\/span><\/b><\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n\\int \\ln x d x=\\ln x \\int d x-\\int\\left\\{\\frac{d}{d x}(\\ln x) \\int d x\\right\\} d x<\/span>\n

 <\/p>\n

=\\ln x \\cdot x-\\int\\left\\{\\frac{1}{x} \\cdot x\\right\\} d x<\/span><\/span><\/p>\n

=x \\ln x-\\int d x<\/span><\/span><\/p>\n

=x \\ln x-x+c<\/span><\/span>\u00a0 \u00a0(Ans)<\/b><\/p>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09ea:\\int e^{x} \\sin 2 x d x<\/span><\/b>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

Example \u2013 4: Determine \\int e^{x} \\sin 2 x d x<\/span><\/b><\/p>\n

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

\u09a7\u09b0\u09bf, I=\\int e^{x} \\sin 2 x d x=\\sin 2 x \\int e^{x} d x-\\int\\left\\{\\frac{d}{d x}(\\sin 2 x) \\int e^{x} d x\\right\\} d x<\/span><\/b><\/span><\/p>\n

=\\sin 2 x \\cdot e^{x}-\\int 2 \\cos 2 x \\cdot e^{x} d x<\/span><\/b><\/p>\n

=e^{x} \\sin 2 x-2\\left[\\cos 2 x \\int e^{x} d x-\\int\\left\\{\\frac{d}{d x}(\\cos 2 x) \\int e^{x} d x\\right\\} d x\\right.<\/span><\/b><\/p>\n

=e^{x} \\sin 2 x-2\\left[\\cos 2 x \\cdot e^{x}-\\int\\left\\{-2 \\sin 2 x \\cdot e^{x}\\right\\} d x\\right.<\/span><\/b><\/p>\n

=e^{x} \\sin 2 x-2 e^{x} \\cos 2 x-4 \\int e^{x} \\sin 2 x d x<\/span><\/b><\/p>\n

\\Rightarrow I=e^{x} \\sin 2 x-2 e^{x} \\cos 2 x-4 I+c<\/span><\/b><\/p>\n

\\Rightarrow 5 I=e^{x}(\\sin 2 x-2 \\cos 2 x)+c<\/span><\/b><\/span><\/p>\n

\\Rightarrow I=\\frac{1}{5} e^{x}(\\sin 2 x-2 \\cos 2 x)+c<\/span><\/b><\/span><\/p>\n

\\therefore \\int e^{x} \\sin 2 x d x=\\frac{1}{5} e^{x}(\\sin 2 x-2 \\cos 2 x)+c<\/span><\/b>\u00a0 \u00a0(Ans)<\/b><\/p>\n

 <\/p>\n

\u09ac\u09bf.\u09a6\u09cd\u09b0. (Note):<\/b> \\int e^{a x} \\cos b x d x=\\frac{e^{a x}}{a^{2}+b^{2}}(a \\cos b x+b \\sin b x)<\/span><\/b><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \\int e^{a x} \\sin b x d x=\\frac{e^{a x}}{a^{2}+b^{2}}(a \\sin b x-b c \\cos b x)<\/span><\/b><\/p>\n

 <\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09eb: <\/b>\\int \\frac{x e^{x}}{(x+1)^{2}} d x<\/span>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n

Example \u2013 5: Determine \\int \\frac{x e^{x}}{(x+1)^{2}} d x<\/span><\/b><\/p>\n

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

 <\/p>\n

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

\u09a7\u09b0\u09bf, f(x)=\\frac{1}{x+1} \\quad \\therefore f^{\\prime}(x)=-\\frac{1}{(x+1)^{2}}<\/span><\/span><\/p>\n\\therefore \\int \\frac{x e^{x}}{(x+1)^{2}} d x=\\int e^{x}\\left\\{f(x)+f^{\\prime}(x)\\right\\} d x=e^{x} f(x)+c<\/span>\n

 <\/p>\n

=\\frac{e^{x}}{x+1}+c<\/span>\u00a0 \u00a0<\/span>\u00a0(Ans)<\/b><\/p>\n

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\u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n