Question

Evaluate the following :$\int \frac{1}{\sqrt{x^2+4x+29}}dx$

Answer 1

$\int \frac{1}{\sqrt{x^2+4x+29}}dx$ can be written as

$\int \frac{1}{\sqrt{x^2+4x+4+25}}dx=\int \frac{1}{\sqrt{{(x+2)}^2+5^2}}dx\{{(x+2)}^2=x^2+4x+4\}$

Aslo we know $\int \frac{1}{\sqrt{x^2+a^2}}dx={\sin }^{-1}h\left(\frac{x}{a}\right)$

here $x=x+2$ and $a=5$

So, $\int \frac{1}{\sqrt{{(x+2)}^2+5^2}}dx={\sin }^{-1}h\left(\frac{x+2}{5}\right)$