# Calculus Integration Problem

Integrate:
$(1+x^{\frac{1}{2}})^{-1} dx$

Solving:

For the integrand $\frac{1}{\sqrt{x}+1}$ substitute $u=\sqrt{x}$ and $du=\frac{1}{(2\sqrt{x})}dx$
$=2\int{\frac{u}{u+1}}du$
For the integrand $\frac{u}{u+1}$, do long division:
$=2\int{(1-\frac{1}{u+1})}du$
Integrate the sum term by term and factor out constants:
$=2\int1du-2\int{\frac{1}{u+1}}du$
For the integrand $\frac{1}{u+1}$ substitute s=u+1 and ds=du
$=2\int1du-2\int\frac{1}{s}ds$
The integral of $\frac{1}{s}$ is log(s)
The integral of 1 is u
$=2u-2\log{(u+1)}$ + constant
Substitute back for $u=\sqrt{x}$ ( =x^1/2)
$=2\sqrt{x} -2\log{(\sqrt{x}+1)}$ + constant
🙂