Question

How do you find the definite integral of `cos^4 x sin x dx` in the interval `[0, pi/3]`?

Answer 1

`31/160`

Explanation:

Knowing that

`d/(dx)cos^n x = -ncos^(n-1)x sin x` and making `n = 5` we have

`-5cos^4 xsin x = d/(dx)cos^5x` so

`cos^4 x sin x = -1/5d/(dx)cos^5 x` and consequently

`int_0^(pi/3) cos^4x sin x dx = -1/5cos^5(pi/3) +1/5cos^5(0)=1/5(1-cos^5(pi/3))=31/160`