What is int cos^3(x) sin^4(x) dx ?

1 Answer
Apr 27, 2018

The integral is equal to (7sin^5x-5sin^7x)/35+C.

Explanation:

Use cos^2x=1-sin^2x, then use a substitution:

color(white)=intcos^3x*sin^4x dx

=intcosx*cos^2x*sin^4x dx

=intcosx*(1-sin^2x)*sin^4x dx

Let u=sinx, so du=cosx dx, and dx=(du)/cosx:

=intcosx*(1-u^2)*u^4\*(du)/cosx

=intcolor(red)cancelcolor(black)cosx*(1-u^2)*u^4\*(du)/color(red)cancelcolor(black)cosx

=int(1-u^2)*u^4 du

=int(u^4-u^6) du

=intu^4 du-intu^6 du

=u^5/5-u^7/7+C

=(7u^5-5u^7)/35+C

=(7sin^5x-5sin^7x)/35+C

That's the integral. Hope this helped!