Mathematics

$$\displaystyle \int{ { e }^{ 3\log { x }  }{ \left( { x }^{ 4 }+1 \right)  }^{ -1 }dx }$$ is equal to ?


ANSWER

$$\dfrac { 1 }{ 4 } \log { \left( { x }^{ 4 }+1 \right) } +C$$


SOLUTION
$$\int e^{3 \log x} (x^{4}+1)^{-1} dx= \int e^{\log x^{3}} (x^{2}+1)^{-1} dx$$
$$= \int \dfrac{x^{3}}{1+x^{4}} dx$$
Let $$14x^{4} =t$$
$$x^{3} dx =dt$$
$$\Rightarrow = \int \dfrac{dt}{4t}$$
$$=\dfrac{1}{4} \log t+c$$
$$=\dfrac{1}{4} \log (1+x^{4})+c$$
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Single Correct Medium Published on 17th 09, 2020
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