Integrate the function: 1 9 x^2+6 x+5 - Vidyadip

Question

Integrate the function: $\frac{1}{9 x^2+6 x+5}$

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Answer

Clearly, $9 x^2+6 x+5=(3 x+1)^2+(2)^2$
$\Rightarrow \int \frac{1}{9 x^2+6 x+5} d x=\int \frac{1}{(3 x+1)^2+(2)^2} d x$
Let $3 x +1= t$
$\Rightarrow 3 dx=dt$
$\therefore \int \frac{1}{(3 x+1)^2+(2)^2} d x=\frac{1}{3} \int \frac{1}{t^2+2^2} d t$
$=\frac{1}{3}\left[\frac{1}{2} \tan ^{-1}\left(\frac{t}{2}\right)\right]+C$
$=\frac{1}{6} \tan ^{-1}\left(\frac{3 x+1}{2}\right)+C$

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