Question

Find:

`inte^x [1/(1+x^2)^(3/2) + x/(sqrt (1+x^2))] dx`

Answer 1

Let we know that 

`I = inte^x [1/(1+x^2)^(3/2) + x/(sqrt (1+x^2))]`

∴ ∫e^x [f(x) + f'(x)] the [e^x f(x) + C] is a solution

`f(x) = x/sqrt (1+x^2)`

`f'(x) = (sqrt(1+x^2) xx (1) - x xx 1/(2sqrt(1+x^2))xx2x)/((sqrt (1+x^2))^2)`

`f'(x) = (sqrt(1+x^2) - x^2/sqrt (1+x^2))/((1+x^2)`

`f'(x) = (1+x^2-x^2)/(1+x^2)^(1/(2+1)) = 1/(1+x^2)^(3/2)`

`I = (xe^x)/sqrt(1+x^2) + C`