Evaluate the following integrals:

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 $t=x^{2}+a^{2}$

$y=x \log \left(x+\sqrt{x^{2}+a^{2}}\right)-\sqrt{x^{2}+a^{2}}+c$