Question

Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`

Answer 1

`(x^3-3x+1)/sqrt(1-x^2)=−(x^2+3x−1+1−1)/sqrt(1−x2)`

`=−(1-x^2+3x−2)/sqrt(1−x2)`

`=(−1−x^2)/sqrt(1−x^2)−(3x−2)/sqrt(1−x^2)`

 

`=−sqrt(1−x2)−(3x−2)/sqrt(1−x2)`

`=>int(x^3-3x+1)/sqrt(1-x^2)dx`

`=int(−sqrt(1−x2)−(3x−2)/sqrt(1−x2))dx`

`=−intsqrt(1−x2)dx−int(3x−2)/sqrt(1−x2)dx`

`=−intsqrt(1−x2)dx−int(3x)/sqrt(1−x2)dx-2int(1)/sqrt(1−x2)dx`

`=−intsqrt(1−x2)dx−int(3x)/sqrt(t)dt-2int(1)/sqrt(1−x2)dx    (Here, t=1−x2.)`

 

`=−[1/2xsqrt(1−x2)+1/2sin^(−1) x]+3/2xx2sqrtt−2cos^(−1) x+C `

`= −1/2xsqrt(1−x2)−1/2sin^(−1) x+3sqrt(1−x2)−2cos^(−1) x+C`