Indefinite integral of the inverse Pythagorean theorem?

Question

So here is my equation:

$$\int{\frac{dx}{(x^2 + d^2)^{1/2}}}$$

Is there any way to solve this? Thanks! Also, $d$ is just a constant.

Answer 1

I hope you won't mind if I use $a$ instead of your $d$, which interferes too much visually with the notation for the derivative.

The simplest approach uses hyperbolic functions. Let $x=a\sinh t$. We end up with $\int dt$, which is $t+C$. But $t=\operatorname{arcsinh}(x/a)$.

Hyperbolic functions are often absent from first calculus courses. Then things get a lot messier. The standard substitution is $x=a\tan \theta$. Then $dx=a\sec^2\theta\,d \theta$. We end up having to find $\int \sec\theta\,d\theta$.

Possibly the integral of $\sec\theta$ is part of your standard list of integrals. If it isn't, we need to find it.

Note that $\sec\theta=\frac{1}{\cos\theta}=\frac{\cos\theta}{1-\sin^2\theta}$. Make the substitution $u=\sin\theta$, and we end up with $\int \frac{1}{1-u^2}\,du$.

But $\frac{1}{1-u^2}=\frac{1}{2}\left(\frac{1}{1-u}+\frac{1}{1+u}\right)$. Now we are at a familiar problem.