Techniques of integration By substitution

Question

Integrate $$\int\frac{1}{1+a^2+x^2}dx$$
where $a$ is constant.

My approach: Substitute $x=\frac{1}{t}$, then I am get an answer in terms of $\arctan x$.

Please tell me whether i am correct or not, since my answer doesn't match with teacher's answer.

Answer 1

Just let $$x=\sqrt{1+a^2}\, t\implies dx=\sqrt{1+a^2}\, dt$$ making $$I=\int \frac{dx}{1+a^2+x^2}=\frac 1{\sqrt{1+a^2}}\int \frac{dt}{1+t^2}$$

Answer 2

Since $1+a^2$ is fixed consider $c=1+a^{2}$. Now put $x=\sqrt{c}\cdot \tan(t)$. Then $dx =\sqrt{c} \cdot \sec^{2}(t) \ dt$ and your integral becomes $$\int \frac{1}{c+c\tan^{2}(t)} \ \sec^{2}(t) dt$$