Integrate the following functions w.r.t. x: e^ 3 x (x^4+1 )^ -1 - Vidyadip

Question

Integrate the following functions w.r.t. x:
$e^{3 \log x} \cdot\left(x^4+1\right)^{-1}$

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Answer

Let $I=e^{3 \log x}\left(x^4+1\right)^{-1} d x$
$=\int \frac{e^{\log x^3}}{x^4+1} d x$
$=\int \frac{x^3}{x^4+1} d x \quad \ldots\left[\because e^{\log N}=N\right]$
$=\frac{1}{4} \int \frac{4 x^3}{x^4+1} d x$
$=\frac{1}{4} \int \frac{\frac{d}{d x}\left(x^4+1\right)}{x^4+1} d x$
$=\frac{1}{4} \log \left|x^4+1\right|+c . \ldots\left[\because \int \frac{f^{\prime}(x)}{f(x)} d x=\log |f(x)|+c\right]$

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