What is the integral of (e^{4x})/(e^{4x)+7} ?

The integral of (e^{4x})/(e^{4x)+7} is 1/4 ln|e^{4x}+7|+C

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integral of (e^{4x})/(e^{4x)+7}

\int \frac{e ^{4 x}}{e ^{4 x} + 7} d x

\frac{1}{4} ln e^{4 x} + 7 + C

Solution steps

\int \frac{e ^{4 x}}{e ^{4 x} + 7} d x

Apply u-substitution

= \int \frac{1}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{4} ln ∣ u ∣

Substitute back

u = e^{4 x} + 7

= \frac{1}{4} ln e^{4 x} + 7

Add a constant to the solution

= \frac{1}{4} ln e^{4 x} + 7 + C

What is the integral of (e^{4x})/(e^{4x)+7} ?
The integral of (e^{4x})/(e^{4x)+7} is 1/4 ln|e^{4x}+7|+C