Question:

Let $I = \int \frac{dx}{\sqrt{x^2 + a^2}}$<br>Put $x = a \tan \theta \Rightarrow dx = a \sec^2 \theta d\theta$<br>$\therefore I = \int \frac{a \sec^2 \theta d\theta}{a \sec \theta} = \int \sec \theta d\theta = \ln |\sec \theta + \tan \theta| + C = \ln |a \sec \theta + a \tan \theta| + C - \ln a = \ln |x + \sqrt{x^2 + a^2}| + C$ [$C - \ln a$ is a constant]<br><b>M-2</b><br>Put $x + \sqrt{x^2 + a^2} = t \Rightarrow (1 + \frac{x}{\sqrt{x^2 + a^2}}) dx = dt \Rightarrow \frac{dx}{\sqrt{x^2 + a^2}} = \frac{dt}{t}$<br>$\therefore I = \int \frac{dx}{\sqrt{x^2 + a^2}} = \int \frac{dt}{t} = \ln |t| + C = \ln |x + \sqrt{x^2 + a^2}| + C$