Question

prove $\int\limits { \frac{1}{\sqrt{a^{2}-x^{2}} \ }} \, dx =sin^{-1} \frac{x}{a}$

Answer 1

Solution:

Let, x = a sinθ so that

dx = a cosθ dθ

Now, $\displaystyle \mathsf{\int \frac{dx}{\sqrt{a^{2}-x^{2}}}}$

= $\displaystyle \mathsf{\int \frac{a\:cos\theta\:d\theta}{\sqrt{a^{2}-a^{2}\:sin^{2}\theta}}}$

= $\displaystyle \mathsf{\int \frac{a\:cos\theta\:d\theta}{\sqrt{a^{2}\:(1-sin^{2}\theta)}}}$

= $\displaystyle \mathsf{\int \frac{a\:cos\theta\:d\theta}{a\:cos\theta}}$

= $\displaystyle \mathsf{\int d\theta}$

= $\displaystyle \mathsf{\theta+C}$

where C is constant of integration

= $\displaystyle \mathsf{sin^{-1}\frac{x}{a}+C}$

which is the required integral.

Hence, proved.