Question

solve: $\int \frac{x+1}{\sqrt{x^2+1}}dx$

Answer 1

$\begin{array}{c}I=\int \frac{x+1}{\sqrt{x^2+1}}dx\\ I=\int \frac{x}{\sqrt{x^2+1}}dx+\int \frac{1}{\sqrt{x^2+1}}dx\\ I=\int \frac{x}{\sqrt{x^2+1}}dx+\int \frac{1}{\sqrt{x^2+1}}dx\\ I=\frac{1}{2}\int \frac{2x}{\sqrt{x^2+1}}dx+\int \frac{1}{\sqrt{x^2+1}}dx\\ I=\sqrt{x^2+1}+\log |x+\sqrt{x^2+1}|+C\end{array}$