Question

Integrate :-

Answer 1

You need to break this apart to start with. This has to become:

\[\int\limits_{}^{}(sinx)(\sin ^{2}x)(\sin ^{2}x)dx\]
From there change the sin squares into cosines using the pythagorean identity and see if you can go from there :P

Answer 2

First, rewrite the function as:

∫sin^4(x) sin(x) dx

Knowing that sin^2(x) = 1 - cos^2(x), you can rewrite the left side as:

∫[1 - cos^2(x)]^2 sin(x) dx

Now you can finish off the rest of this with u-substitution:

u = cos(x)
du = -sin(x) dx

This leaves you with:

-∫(1 - u^2)^2 du

Now do the squaring:

-∫(1 - 2u^2 + u^4) du

Integrate with respect to u:

-u + (2/3)u^3 - (1/5)u^5 + C

Now substitute back in u = cos(x) to get your final answer:

-cos(x) + (2/3)cos^3(x) - (1/5)cos^5(x) + C