Evaluate the integral of 1/(a²-x²)^1/2

by RoRi

November 29, 2015

Evaluate the following integral for $a \neq 0$ ,

$\int \frac{dx}{\sqrt{a^2 - x^2}}.$

To evaluate the integral, first we pull out a $\frac{1}{|a|}$ ,

$\int \frac{dx}{\sqrt{a^2 - x^2}} = \int \frac{dx}{|a| \sqrt{1 - \left( \frac{x}{a}\right)^2}}.$

Then, we make the substitution $u = \frac{x}{|a|}$ and $du = \frac{dx}{|a|}$ . This gives us

$\begin{align*} \int \frac{dx}{\sqrt{a^2-x^2}} &= \int \frac{du}{\sqrt{1-u^2}} \\ &= \arcsin u + C \\ &= \arcsin \frac{x}{|a|} + C. \end{align*}$

Stumbling Robot