Question

What is the integral of `int sin^5x*cos^7x`?

Answer 1

`-(1/8cos^8 x-2/10 cos^10 x + 1/12 cos^12 x) + C`

Explanation:

`sin^5x*cos^7x = sin^4x cos^7x sinx = (1-cos^2 x)^2cos^7x sin x`
`=(1-2cos^2x+cos^4 x)cos^7 x sin x`
`=(cos^7x-2 cos^9 x+cos^11 x)sinx` so

`int sin^5x*cos^7x dx = int (cos^7x-2 cos^9 x+cos^11 x)sinx dx`
`=-(1/8cos^8 x-2/10 cos^10 x + 1/12 cos^12 x) + C`

or simplifying

`-1/480(27 - 28 cos(2 x) + 5 cos(4 x)) cos^8x`