How to integrate sin^4(x)cos(x)?

To integrate sin^4(x)cos(x), use the substitution u = sin(x) and simplify using trigonometric identities.

To integrate sin^4(x)cos(x), start by using the substitution u = sin(x). This gives:

∫sin^4(x)cos(x) dx = ∫u^4 du

Next, simplify the integrand using trigonometric identities. Recall that sin^2(x) + cos^2(x) = 1, so we can write:

sin^4(x) = (sin^2(x))^2 = (1 - cos^2(x))^2

Substituting this into the original integral and using the identity cos(2x) = 1 - 2sin^2(x), we get:

∫sin^4(x)cos(x) dx = ∫(1 - cos^2(x))^2 cos(x) dx
= ∫(1 - 2sin^2(x) + sin^4(x))cos(x) dx
= ∫cos(x) dx - 2∫sin^2(x)cos(x) dx + ∫sin^4(x)cos(x) dx
= sin(x) - 2∫(1 - cos^2(x)) d(cos(x)) + ∫(1 - cos^2(x))^2 d(cos(x))
= sin(x) - 2(cos(x) - 1/3cos^3(x)) + 1/5(cos(x) - 1/3cos^3(x))^2 + C

Therefore, the final answer is:

∫sin^4(x)cos(x) dx = sin(x) - 2(cos(x) - 1/3cos^3(x)) + 1/5(cos(x) - 1/3cos^3(x))^2 + C