`int sqrt(cos x(cos^(3) x + sin 2x))dx = (1- sinx)/(k)sqrt(1+2 sin x-sin^(2)x) + sin^(-1)((1-sinx)/(sqrt(2)))+c` then 3k =

Answer

6

Explanation

`int sqrt(cos^(4) x+ 2 sin x cos^(2)x)dx`
`= int cos xsqrt(cos^(2)x + 2 sin x dx)`
`= int cos xsqrt(1+2sin x - sin^(2)x)dx = int sqrt(1+2t-t^(2))dx = intsqrt(1-[t^(2)-2t+1-2])dt`
`= int sqrt(2-(1-t)^(2)) dt=(1-t)/(2) sqrt(1+2t-t^(2))+(2)/(2) sin^(-1)((1-t)/(sqrt(2)))`
`=(1- sin x)/(2)sqrt(1+ 2sin x - sin^(2)x) + sin^(-1)((1-sin x)/(sqrt(2)))+c`