Question

Solve: $\int \frac{xdx}{\sqrt{x}+1}$

Answer 1

$\begin{array}{c}\int \frac{xdx}{\sqrt{x}+1}\\ \sqrt{x}=t-1\\ dx=2\left(t-1\right)dt\\ 2\int \frac{{\left(t-1\right)}^{3}dt}{t}\\ =2\int \frac{t^3-3t^2+3t-1dt}{t}\\ =2\int t^2-3t+3-\frac{1}{t}dt\\ =2\left[\frac{t^3}{3}-\frac{3t^2}{2}+3t-\log t\right]+C\\ =2\left[\frac{{\left(\sqrt{x}+1\right)}^3}{3}-\frac{3{\left(\sqrt{x}+1\right)}^2}{2}+3\left(\sqrt{x}+1\right)-\log \left(\sqrt{x}+1\right)\right]+C\end{array}$