Find `intxsqrt(1+x-x^2)dx`

Explanation

Let `x=lambda. d/(dx)(1+x-x^(2))+mu`
`rArr x=lambda(1-2x)+mu`
Comparing the coefficients of like powers of x, we get
`1=-2lambda` and `lambda+mu=0 rArr lambda=-1/2` and `mu=1/2 therefore x=-1/2(1-2x)+1/2`
So, `intxsqrt(1+x-x^(2))dx`
`=int{-1/2(1-2x)+1/2}sqrt(1+x-x^(2)dx) = -1/2int(1-2x)sqrt(1+x-x^(2)dx+1/2intsqrt(1+x-x^(2))dx`
`=-1/2intsqrt(1+x-x^(2))d(1+x-x^(2))+1/2intsqrt((sqrt(5)/2)^(2)-(x-1/2)^(2))dx`
`=-1/3(1+x-x^(2))^(3//2)+1/2[1/2(x-1/2)sqrt((sqrt(5)/(2))^(2)-(x-1/2)^(2))+1/2(sqrt(5)/(2))^(2)sin^(-1)(x-1//2)/(sqrt(5)//2)]+C`
`=-1/3(1+x-x^(2))^(3//2)+1/2[(x-1/2)sqrt(1+x-x^(2))+5/8sin^(-1)(2x-1)/(sqrt(5))]+C`