Question

How do you evaluate the definite integral `int sin^(5)x * cos^(20)x dx` from [0,pi/2]?

Answer 1

`1/21-2/23+1/25`

Explanation:

`int_0^(pi/2) sin^(5)x * cos^(20)x dx =int_0^(pi/2) (1-cos^2x)^2cos^20 x sin x dx`
`=int_0^(pi/2) (1-2cos^2x+cos^4x)cos^20 x sin xdx =`
`=int_0^(pi/2)(cos^20x-2cos^22 x+cos^24x)sinx dx =`
`=-1/21(0-1)+2/23(0-1)-1/25(0-1)=`
`=1/21-2/23+1/25`