If $\int \frac{x^{e - 1} + e^{x - 1}}{x^{e} + e^{x}} d x = k ln | x^{e} + e^{x} | + C,$ then the value of $4 ln k + 7$ is equal to
(where $C$ is constant of integration)

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Solution

$I = \int \frac{x^{e - 1} + e^{x - 1}}{x^{e} + e^{x}} d x = \frac{1}{e} \int \frac{e x^{e - 1} + e^{x}}{x^{e} + e^{x}} d x$
putting $x^{e} + e^{x} = t$
$\Rightarrow (e x^{e - 1} + e^{x}) d x = d t ∴ I = \frac{1}{e} \int \frac{1}{t} d t = \frac{1}{e} ln | t | + C = \frac{1}{e} ln | x^{e} + e^{x} | + C$
$\Rightarrow k = \frac{1}{e} \Rightarrow 4 ln (\frac{1}{e}) + 7 = - 4 + 7 = 3$

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