What is the integral of (e^{7x})/(e^{7x)+7} ?

The integral of (e^{7x})/(e^{7x)+7} is 1/7 ln|e^{7x}+7|+C

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integral of (e^{7x})/(e^{7x)+7}

\int \frac{e ^{7 x}}{e ^{7 x} + 7} d x

\frac{1}{7} ln e^{7 x} + 7 + C

Solution steps

\int \frac{e ^{7 x}}{e ^{7 x} + 7} d x

Apply u-substitution

= \int \frac{1}{7 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{7} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{7} ln ∣ u ∣

Substitute back

u = e^{7 x} + 7

= \frac{1}{7} ln e^{7 x} + 7

Add a constant to the solution

= \frac{1}{7} ln e^{7 x} + 7 + C

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What is the integral of (e^{7x})/(e^{7x)+7} ?
The integral of (e^{7x})/(e^{7x)+7} is 1/7 ln|e^{7x}+7|+C