Question

How do you find the indefinite integral of ∫ dx / √(x+1) + √(x+1)^3 ?

Answer 1

`I=2arctan(sqrt(x+1))+C`

Explanation:

We want to solve

`I=int1/(sqrt(x)+sqrt((x+1)^3))dx`

Rewrite the integrand as

`I=int1/((x+1)^(1/2)+(x+1)^(3/2))dx`

Make a substitution `u=x+1=>du=dx`

`I=int1/(u^(1/2)+u^(3/2))du`

Make a substitution `s=sqrt(u)=>ds=1/(2s)du`

`I=int(2s)/(s+s^3)ds`

`color(white)(I)=2int1/(1+s^2)ds`

`color(white)(I)=2arctan(s)+C`

Substitute back `s=sqrt(u)` and `u=x+1`

`I=2arctan(sqrt(x+1))+C`