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

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 $$u=\frac{x}{\sqrt{5}}$$?

• Yes, that's the good way to see it – charmd May 20 at 15:51
• *Derivative of $\arctan(u)$ – Shubham Johri May 20 at 15:57

$$\int \frac{1}{x^2 + a^2} dx$$
$$= \int\frac{1}{a^2} \frac{1}{\left(\frac{x}{a}\right)^2 +1 }dx$$
Now letting $$\frac{x}{a} = t$$, we arrive at known integral.
Write your integrand of the form $$\frac{5}{2\left(\left(\frac{x}{\sqrt{2}}\right)^2+1\right)}$$ and substitute $$t=\frac{x}{\sqrt{2}}$$