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Get Answers to all your Questions
Q: Integrate the function $e^{3 \log x}\left(x^4+1\right)^{-1}$.
Answers (1)
Given the function to be integrated as $e^{3 \log x}\left(x^4+1\right)^{-1}$
$=e^{\log x^3}\left(x^4+1\right)^{-1}=\frac{x^3}{x^4+1}$
$I=\int e^{3 \log x}\left(x^4+1\right)^{-1}$
Let $x^4=t \Longrightarrow 4 x^3 d x=d t$
$I=\int e^{3 \log x}\left(x^4+1\right)^{-1}=\int \frac{x^3}{x^4+1}$
$=\frac{1}{4} \cdot \int \frac{1}{\mathrm{t}+1} \cdot \mathrm{dt}$
$=\frac{1}{4} \log (\mathrm{t}+1)+\mathrm{C}$
$\Longrightarrow I=\frac{1}{4} \log \left(x^4+1\right)+C$
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