what is the integral of sin(x)cos^5(x)?

Answer 1

note you have cos^5 and you have a lone sin(x), which works out perfectly for u substitution

∫sin(x) cos^5(x) dx

Using u substitution
u = cos(x)
du = -sin(x) dx
dx = du/-sin(x)

Substituting in you obtain
-∫u^5 du

Integrating
-u^6/6 +C

reverting back to x
-cos^6(x)/6 +C

Answer 2

Since the derivative of cos(x) is -sin(x), the integral is

- cos^6(x)/6 + C

Answer 3

-sin^2x5cos^4x+cos^6x
=c0s^4x(cos^2x-sin^2x)
c0s^4x(2cos^2x-1)