Integrate the function e^ 3 x (x^ 4 +1 )^ -1 - Vidyadip

Question

Integrate the function $e^{3 \log x}\left(x^{4}+1\right)^{-1}$

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Answer

Let $\mathrm{I}=\mathrm{e}^{3 \log \mathrm{x}}\left(\mathrm{x}^{4}+1\right)^{-1}$
$=e^{\log x^{3}}\left(x^{4}+1\right)^{-1}=\frac{x^{3}}{x^{4}+1}$
Let $x^4 = t$
$\Rightarrow 4x^3 dx = dt$
$\Rightarrow x^3 dx = \frac{dt}{4}$
$\Rightarrow \int e^{3 \log x}\left(x^{4}+1\right)^{-1}=\int \frac{x^{3}}{x^{4}+1} d x$
$= \int \frac{1}{t+1} \cdot \frac{d t}{4}$
$= \frac{1}{4} \cdot \int \frac{1}{t+1} \cdot d t$
$= \frac{1}{4} \log (t+1)+C$
$\Rightarrow I=\frac{1}{4} \log \left(x^{4}+1\right)+c$

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