Question

Evaluate $\int \frac{1}{\sqrt{25+9x^2}}dx$

Answer 1

We have,

$I=\int \frac{1}{\sqrt{25+9x^2}}dx$

$I=\frac{1}{3}\int \frac{1}{\sqrt{{\left(\frac{5}{3}\right)}^2+x^2}}dx$

So, applying formula,

$\int \frac{1}{\sqrt{a^2+x^2}}dx=\ln [x+\sqrt{x^2+a^2}]$

Therefore,

$I=\frac{1}{3}\ln [x+\sqrt{x^2+{\left(\frac{5}{3}\right)}^2}]+C$

Hence, this is the answer.