Evaluate the following integrals:

Let $\mathrm{bx}=\mathrm{t}$ then $\mathrm{dt}=\mathrm{bdx}$ or $\mathrm{dx}=\frac{\mathrm{dt}}{\mathrm{b}}$

Hence, $\int \frac{1}{\sqrt{a^{2}-b^{2} x^{2}}} d x=\frac{1}{b} \int \frac{1}{\sqrt{\left(a^{2}-t^{2}\right)}} d t$

$=\frac{1}{b} \int \sin ^{-1}\left(\frac{t}{a}\right)+c\left\{\right.$ since $\left.\int \frac{1}{\sqrt{a^{2}-x^{2}}} d x=\sin ^{-1}\left(\frac{x}{a}\right)+c\right\}$

Put $t=b x$

$=\frac{1}{b} \int \sin ^{-1}\left(\frac{b x}{a}\right)+c$