# Solve a problem on integration

$$\int x\sqrt{x+x^2}\,dx = ?$$

Can you solve this problem? I have tried this after a challenge. 😊

#### Solutions Collecting From Web of "Solve a problem on integration"

Hint:

Set $x^2+x=u$, we have
$$\int x\sqrt{x+x^2}\,dx =\frac 12 \int (2x+1)\sqrt{x+x^2}\,dx-\frac 12 \int \sqrt{-\frac 14+\left(x+\frac 12\right)^2}\,dx$$

Let $x+\frac12=\frac12\cosh(u)$, then $\sqrt{x+x^2}=\frac12\sinh(u)$ and $\mathrm{d}x=\frac12\sinh(u)\,\mathrm{d}u$. Furthermore, $\sinh^2(u)=\frac{\cosh(2u)-1}2$
\begin{align} \int x\sqrt{x+x^2}\,\mathrm{d}x &=\frac18\int(\cosh(u)-1)\sinh^2(u)\,\mathrm{d}u\\ &=\frac1{24}\sinh^3(u)-\frac1{32}\sinh(2u)+\frac1{16}u+C\\ &=\frac1{24}\sinh^3(u)-\frac1{16}\sinh(u)\cosh(u)+\frac1{16}u+C\\ &=\frac13\sqrt{x+x^2}^{\,3}-\frac18(2x+1)\sqrt{x+x^2}+\frac1{16}\cosh^{-1}(2x+1)+C \end{align}

Hint: You can do that easily using Euler’s substitution.