Question:

Let $I = \int \sqrt{x^2 - a^2} dx$<br>Taking constant function 1 as the second function and integration by parts, we have<br>$$I = x\sqrt{x^2 - a^2} - \int x \cdot \frac{1}{2} \frac{2x}{\sqrt{x^2 - a^2}} dx$$<br>$$= x\sqrt{x^2 - a^2} - \int \frac{x^2}{\sqrt{x^2 - a^2}} dx = x\sqrt{x^2 - a^2} - \int \frac{x^2 - a^2 + a^2}{\sqrt{x^2 - a^2}} dx$$<br>$$= x\sqrt{x^2 - a^2} - \int \sqrt{x^2 - a^2} dx - a^2 \int \frac{dx}{\sqrt{x^2 - a^2}}$$<br>$$= x\sqrt{x^2 - a^2} - I - a^2 \int \frac{dx}{\sqrt{x^2 - a^2}}$$<br>or $$2I = x\sqrt{x^2 - a^2} - a^2 \int \frac{dx}{\sqrt{x^2 - a^2}}$$<br>or $$I = \int \sqrt{x^2 - a^2} dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log |x + \sqrt{x^2 - a^2}| + C$$