Evalute : 1 4 x^2-20 x+17 d x - Vidyadip

Question

Evalute : $\int \frac{1}{4 x^2-20 x+17} d x$

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Answer

$
\begin{aligned}
& \int \frac{1}{4 x^2-20 x+17} d x \\
= & \frac{1}{4} \int \frac{1}{x^2-5 x+\frac{17}{4}} d x \\
= & \frac{1}{4} \int \frac{1}{\left(x^2-5 x+\frac{25}{4}\right)-\frac{25}{4}+\frac{17}{4}} d x \\
& =\frac{1}{4} \int \frac{1}{\left(x-\frac{5}{2}\right)^2-(\sqrt{2})^2} d x
\end{aligned}
$
$
\begin{aligned}
& =\frac{1}{4} \times \frac{1}{2 \sqrt{2}} \log \left|\frac{x-\frac{5}{2}-\sqrt{2}}{x-\frac{5}{2}+\sqrt{2}}\right|+c \\
& =\frac{1}{8 \sqrt{2}} \log \left|\frac{2 x-5-2 \sqrt{2}}{2 x-5+2 \sqrt{2}}\right|+c .
\end{aligned}
$

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