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Asad0000Best ResponseYou've already chosen the best response.1
split the cos^3x into cosx and cos^2x then use the trigonometric substitution. Replace cos^2x by (1  sin^2x). Now you can use some integrate it.
 one year ago

sha0403Best ResponseYou've already chosen the best response.0
but i still did not get sin x  1/3 sin^3 (x)+c... bcoz the question ask to prove..
 one year ago

Asad0000Best ResponseYou've already chosen the best response.1
Look. When you did the substitutions as I told you above, you should end up with the following integral:\[\int\limits_{}^{} [(1  \sin^{2}(x) )\cos(x) ]dx\]
 one year ago

RadEnBest ResponseYou've already chosen the best response.0
int (1sin^2 x)cosx dx = int cosx dx  int sin^2 x cosx dx for int cosx dx = .... and to find int sin^2 x cosx dx, use int by sub, let u=sinx du = .... continue it
 one year ago

sha0403Best ResponseYou've already chosen the best response.0
ok the i get integration of (cos x  u^2) du.. how to solve integration of cos x in term of u?
 one year ago

Asad0000Best ResponseYou've already chosen the best response.1
if u = sin(x), then du = cos(x)dx. So replace the sin^2(x) in the original integral by u^2 and cos(x)dx by du. You will have a really trivial integral.
 one year ago

Eng.MidoBest ResponseYou've already chosen the best response.1
\[\int\limits(\cos ^{3}{x})=\int\limits(cosx*\cos^{2}{x})=\int\limits(cosx(1\sin ^{2}{x})=\int\limits(cosx)(sinx)^2(cosx)=sinx1/3(\sin ^{3}{x})+\]
 one year ago

Eng.MidoBest ResponseYou've already chosen the best response.1
a proof for my answer: cos^3x can be written as cos^2x*cosx. Then, we all know that cos^2x can be written as 1sin^2x. then we solve the bracket. then we'll find (sinx)^2*cosx , this remind us of the integration rule where [f(x)]^n*f'(x) = ([f(x)]^n+1/n+1)+c
 one year ago

sha0403Best ResponseYou've already chosen the best response.0
but how to solve the integration of cos xu^2 du?
 one year ago

Eng.MidoBest ResponseYou've already chosen the best response.1
is this ur question ?
 one year ago

sha0403Best ResponseYou've already chosen the best response.0
no.. like this integration of (cos x)(u^2) du
 one year ago

sirm3dBest ResponseYou've already chosen the best response.0
@sha0403\[\Large \int\limits (1 \sin^2 x)(\cos x dx)\]let \(u=\sin x\), \(du = \cos x dx\)
 one year ago

sha0403Best ResponseYou've already chosen the best response.0
yes but how ti integrate cos x in term of du?
 one year ago

Eng.MidoBest ResponseYou've already chosen the best response.1
use the substitution method.
 one year ago

sha0403Best ResponseYou've already chosen the best response.0
ok i get it now.. thank u..sorry i was too slow..
 one year ago

Eng.MidoBest ResponseYou've already chosen the best response.1
\[\cos^2x*cosx dx=\cos^3x \]
 one year ago
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