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

1 Answer
Apr 16, 2018

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