How do I evaluate the indefinite integral intsin^3(x)*cos^2(x)dx∫sin3(x)⋅cos2(x)dx ?
1 Answer
Sep 21, 2014
The answer is
The trick with sinusoidal powers is to use identities so that you can have
In this case, it is easier to get
int sin^3x*cos^2x dx∫sin3x⋅cos2xdx
=int sin x(1-cos^2x)cos^2x dx=∫sinx(1−cos2x)cos2xdx
=int sin x(cos^2x-cos^4x)dx=∫sinx(cos2x−cos4x)dx
=int sin x cos^2xdx-int sin x cos^4x dx=∫sinxcos2xdx−∫sinxcos4xdx
Now it is a matter of using substitution:
u=cos xu=cosx
du = -sin x dxdu=−sinxdx
int sin x cos^2xdx-int sin x cos^4x dx∫sinxcos2xdx−∫sinxcos4xdx
=int -u^2 du+ int u^4 du=∫−u2du+∫u4du
=-(u^3)/3+(u^5)/5+C=−u33+u55+C
=-(cos^3x)/3+(cos^5x)/5+C=−cos3x3+cos5x5+C