Evalute : x^3 16 x^8-25 d x - Vidyadip

Question

Evalute : $\int \frac{x^3}{16 x^8-25} d x$

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Answer

Let $I=\int \frac{x^3}{16 x^8-25} d x$
Put $x^4=t \quad \therefore 4 x^3 d x=d t$
$
\therefore x^3 d x=\frac{d t}{4}
$
$
\begin{aligned}
\therefore I & =\int \frac{1}{16 t^2-25} \cdot \frac{d t}{4} \\
& =\frac{1}{4} \times \frac{1}{16} \int \frac{1}{t^2-\frac{25}{16}} d t \\
& =\frac{1}{64} \int \frac{1}{t^2-\left(\frac{5}{4}\right)^2} d t
\end{aligned}
$
$
\begin{aligned}
& =\frac{1}{64} \times \frac{1}{2 \times \frac{5}{4}} \log \left|\frac{t-\frac{5}{4}}{t+\frac{5}{4}}\right|+c \\
& =\frac{1}{160} \log \left|\frac{4 t-5}{4 t+5}\right|+c \\
& =\frac{1}{160} \log \left|\frac{4 x^4-5}{4 x^4+5}\right|+c .
\end{aligned}
$

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