Question

Which of the following represents the integral  $\displaystyle{\int \frac{x}{\sqrt{1+x^2}}dx}$

1) $\dfrac{x^2}{\sqrt{1+x}}+C$

2) $\ln{(1+x^2)}+C$

3) $\sqrt{1+x^2}+C$

4) $\dfrac{x^2}{\sqrt{1+x^2}}+C$

Answer 1

We rewrite the integral as: $\int\frac{x}{\sqrt{1+x^2}}dx=\frac{1}{2}\int\frac{d(1+x^2)}{\sqrt{1+x^2}}$ and make the change of variables: $z=1+x^2$. We receive: $\int\frac{x}{\sqrt{1+x^2}}dx=\frac{1}{2}\int\frac{d(1+x^2)}{\sqrt{1+x^2}}=\frac{1}{2}\int\frac{dz}{\sqrt{z}}$

We integrate the latter and receive: $\frac{1}{2}\int\frac{dz}{\sqrt{z}}=\sqrt{z}+C=\sqrt{1+x^2}+C,C\in{\mathbb{R}}$.