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If $\int \frac{d x}{1 + e^{x}} = x + f (x) + C,$ then $f (x)$ is equal to

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detailed solution

Correct option is B

Let 1+ex=t⇒ex=t−1⇒x=log⁡(t−1) so dx=1t−1dtThus I=∫dx1+ex=∫dtt(t−1)=∫1t−1−1tdt=log⁡(t−1)−log⁡t+C=log⁡ex−log⁡1+ex+C=x−log⁡1+ex+C=x+log⁡11+ex+C

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If ∫dx1+ex=x+f(x)+C,then f(x) is equal to

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If $\int \frac{d x}{1 + e^{x}} = x + f (x) + C,$ then $f (x)$ is equal to