What is the integral of (8e^{2x})/(3+e^x) ?

The integral of (8e^{2x})/(3+e^x) is 8(3+e^x-3ln|3+e^x|)+C

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integral of (8e^{2x})/(3+e^x)

\int \frac{8 e ^{2 x}}{3 + e ^{x}} d x

8 (3 + e^{x} - 3 ln ∣ 3 + e^{x} ∣) + C

Solution steps

\int \frac{8 e ^{2 x}}{3 + e ^{x}} d x

Take the constant out:

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

= 8 \cdot \int \frac{e ^{2 x}}{3 + e ^{x}} d x

Apply u-substitution

: \int \frac{u - 3}{u} d u

= 8 \cdot \int \frac{u - 3}{u} d u

Expand

\frac{u - 3}{u} : 1 - \frac{3}{u}

= 8 \cdot \int 1 - \frac{3}{u} d u

Apply the Sum Rule:

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

= 8 (\int 1 d u - \int \frac{3}{u} d u)

\int 1 d u = u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= 8 (u - 3 ln ∣ u ∣)

Substitute back

u = 3 + e^{x}

= 8 (3 + e^{x} - 3 ln ∣ 3 + e^{x} ∣)

Add a constant to the solution

= 8 (3 + e^{x} - 3 ln ∣ 3 + e^{x} ∣) + C

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What is the integral of (8e^{2x})/(3+e^x) ?
The integral of (8e^{2x})/(3+e^x) is 8(3+e^x-3ln|3+e^x|)+C