The integral of $5/\left(x^2+2\right)$

Question

I have to calculate a integral for following equation: $\frac{5}{x^2+2}$. On the integral calculator they show that it must be solved by substitution and the substitution must be $u=\frac{x}{\sqrt{2}}$ , Is it because we need to get a derivative of tan function $\frac{1}{u^2+1}$? If the denominator would be let's say $x^2+5$ would we choose $u=\frac{x}{\sqrt{5}}$?

Answer 1

$$\int \frac{1}{x^2 + a^2} dx$$

$$ = \int\frac{1}{a^2} \frac{1}{\left(\frac{x}{a}\right)^2 +1 }dx$$

Now letting $\frac{x}{a} = t$, we arrive at known integral.

Answer 2

Write your integrand of the form $$\frac{5}{2\left(\left(\frac{x}{\sqrt{2}}\right)^2+1\right)}$$ and substitute $$t=\frac{x}{\sqrt{2}}$$