Integration substituion

Question

I need to use a suitable substitution to show $$\int {dx \over \sqrt{a^2+x^2}} = \text{arcsinh} \left(\frac{x}{a}\right)+C$$

but I am not sure what substitution to use. Any help would be great.

Thanks!

Answer 1

Hint

If you know that $$\cosh ^2u-\sinh^2u=1,$$ then, a suitable substitution is $$x=a\sinh(u).$$

Added

Suppose $a>0$. If $x=a\sinh(u)$, then $\mathrm d x=a\cosh(u)\mathrm d u$. Therefore,

$$\int^t\frac{1}{\sqrt{a^2+x^2}}\mathrm d x\underset{x=a\sinh(u)}{=}\int^{\text{arcsinh}(t)}\frac{a\cosh(u)}{\sqrt{a^2(1+\sinh^2(u))}}\mathrm d u=...$$

Answer 2

If you put $a\tan(u)=x$ the integral becomes $$\int\sec\left(u\right)du$$ which is classical.