What is the integral of 1/(e^{5x)+1} ?

The integral of 1/(e^{5x)+1} is -1/5 ln|e^{5x}+1|+x+C

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integral of 1/(e^{5x)+1}

\int \frac{1}{e ^{5 x} + 1} d x

- \frac{1}{5} ln e^{5 x} + 1 + x + C

Solution steps

\int \frac{1}{e ^{5 x} + 1} d x

\frac{1}{e ^{5 x} + 1} = - \frac{e ^{5 x}}{e ^{5 x} + 1} + 1

= \int - \frac{e ^{5 x}}{e ^{5 x} + 1} + 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{e ^{5 x}}{e ^{5 x} + 1} d x + \int 1 d x

\int \frac{e ^{5 x}}{e ^{5 x} + 1} d x = \frac{1}{5} ln e^{5 x} + 1

\int 1 d x = x

= - \frac{1}{5} ln e^{5 x} + 1 + x

Add a constant to the solution

= - \frac{1}{5} ln e^{5 x} + 1 + x + C

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What is the integral of 1/(e^{5x)+1} ?
The integral of 1/(e^{5x)+1} is -1/5 ln|e^{5x}+1|+x+C