Integral substitution, convert integral

Question

I need to find a good substitution to convert integral $\int \frac{1}{t^2+a^2}dt$ to integral $\int \frac{1}{x^2+1}dx$. Can anybody please help me? I don't know the method to use to I approach it. I know that: $\int \frac{dx}{x^2+a^2}=\frac{1}{a}arctg\frac{x}{a}+c$. The problem is to get rid of the $a$.

Answer 1

$t^2 + a^2 = a^2((t/a)^2 + 1)$

so $x=t/a$ is a sensible substitution, with $a\ dx = dt$

$\int \frac{1}{t^2+a^2}dt = \int \frac{1}{a^2(x^2+1)}a\ dx = \frac1a\int\frac{1}{x^2+1}dx$

Can you take it from there?