Evalute : x 4 x^4-20 x^2-3 d x - Vidyadip

Question

Evalute : $\int \frac{x}{4 x^4-20 x^2-3} d x$

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Answer

Let $I=\int \frac{x}{4 x^4-20 x^2-3} d x$
Put $x^2=t \quad \therefore 2 x d x=d t$
$\therefore x d x=\frac{d t}{2}$
$
\therefore I=\int \frac{1}{4 t^2-20 t-3} \cdot \frac{d t}{2}
$
$=\frac{1}{2} \times \frac{1}{4} \int \frac{1}{t^2-5 t-\frac{3}{4}} d t$
$=\frac{1}{8} \int \frac{1}{\left(t^2-5 t+\frac{25}{4}\right)-\frac{25}{4}-\frac{3}{4}} d t$
$=\frac{1}{8} \int \frac{1}{\left(t-\frac{5}{2}\right)^2-(\sqrt{7})^2} d t$
$=\frac{1}{8} \times \frac{1}{2 \sqrt{7}} \log \left|\frac{t-\frac{5}{2}-\sqrt{7}}{t-\frac{5}{2}+\sqrt{7}}\right|+c$
$=\frac{1}{16 \sqrt{7}} \log \left|\frac{2 t-5-2 \sqrt{7}}{2 t-5+2 \sqrt{7}}\right|+c$
$=\frac{1}{16 \sqrt{7}} \log \left|\frac{2 x^2-5-2 \sqrt{7}}{2 x^2-5+2 \sqrt{7}}\right|+c$.

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