Mathematics

Integrate $$\int {\frac{e^{x-1}+x^{e-1}}{e^{x}+x^e}dx}$$.


SOLUTION
Let $$I=\int { \dfrac { { e }^{ x-1 }+{ x }^{ e-1 } }{ { e }^{ x }+{ x }^{ e } }  } dx$$
$$I=\dfrac { 1 }{ e } \int { \dfrac { { e }^{ x }+{ ex }^{ e-1 } }{ { e }^{ x }+{ x }^{ e } }  } dx$$
Let $$\left( { e }^{ x }+{ x }^{ e } \right) =z$$
So,  $$\left( { e }^{ x }+{ ex }^{ e-1 } \right) dx=dz$$
So,  $$I=\dfrac { 1 }{ e } \int { \dfrac { 1 }{ z } dz } =\dfrac { 1 }{ e } log\left| z \right| +C$$
$$I=\dfrac { 1 }{ e } \log\left| { e }^{ x }+{ x }^{ e } \right| +C$$
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Subjective Medium Published on 17th 09, 2020
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