What's the integral of cos(x)sin^3(x)?

The integral of cos(x)sin^3(x) is (-1/4)cos^4(x) + C.

To solve this integral, we can use the substitution u = sin(x), du = cos(x)dx. Then the integral becomes:

∫cos(x)sin^3(x)dx = ∫u^3du

Integrating u^3, we get:

(1/4)u^4 + C = (1/4)sin^4(x) + C

However, we need to substitute back in for u to get the final answer:

(1/4)sin^4(x) + C = (1/4)(1-cos^2(x))^2 + C

Expanding the square and simplifying, we get:

(1/4)(1-2cos^2(x)+cos^4(x)) + C

Simplifying further, we get:

(-1/4)cos^4(x) + (1/2)cos^2(x) + C

Finally, we can simplify this to:

(-1/4)cos^4(x) + C

Therefore, the integral of cos(x)sin^3(x) is (-1/4)cos^4(x) + C.