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Question:

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

Solution:

$e^{3 \log x}\left(x^{4}+1\right)^{-1}=e^{\log x^{3}}\left(x^{4}+1\right)^{-1}=\frac{x^{3}}{\left(x^{4}+1\right)}$

Let $x^{4}+1=t \Rightarrow 4 x^{3} d x=d t$

$\Rightarrow \int e^{3 \log x}\left(x^{4}+1\right)^{-1} d x=\int \frac{x^{3}}{\left(x^{4}+1\right)} d x$

$=\frac{1}{4} \int \frac{d t}{t}$

$=\frac{1}{4} \log |t|+\mathrm{C}$

$=\frac{1}{4} \log \left|x^{4}+1\right|+\mathrm{C}$

$=\frac{1}{4} \log \left(x^{4}+1\right)+\mathrm{C}$

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