Evaluate $\int \log \left(x+\sqrt{x^{2}+a^{2}}\right) d x$
Use method of integration by parts
$y=\log \left(x+\sqrt{x^{2}+a^{2}}\right) \int d x-\int \frac{d}{d x} \log \left(x+\sqrt{x^{2}+a^{2}}\right)\left(\int d x\right) d x$
$y=x \log \left(x+\sqrt{x^{2}+a^{2}}\right)-\int \frac{1+\frac{2 x}{2 \sqrt{x^{2}+a^{2}}}}{x+\sqrt{x^{2}+a^{2}}} x d x$
$y=x \log \left(x+\sqrt{x^{2}+a^{2}}\right)-\int \frac{x}{\sqrt{x^{2}+a^{2}}} d x$
Let, $\mathrm{x}^{2}+\mathrm{a}^{2}=\mathrm{t}$
Differentiating both side with respect to t
$2 x \frac{d x}{d t}=1 \Rightarrow x d x=\frac{d t}{2}$
$y=x \log \left(x+\sqrt{x^{2}+a^{2}}\right)-\frac{1}{2} \int \frac{1}{\sqrt{t}} d t$
$y=x \log \left(x+\sqrt{x^{2}+a^{2}}\right)-\sqrt{t}+c$
Again, put $t=x^{2}+a^{2}$
$y=x \log \left(x+\sqrt{x^{2}+a^{2}}\right)-\sqrt{x^{2}+a^{2}}+c$