What is the integral of (e^{2x})/(5+e^x) ?

The integral of (e^{2x})/(5+e^x) is 5+e^x-5ln|5+e^x|+C

integral of (e^{2x})/(5+e^x)

\int \frac{e ^{2 x}}{5 + e ^{x}} d x

5 + e^{x} - 5 ln ∣ 5 + e^{x} ∣ + C

Solution steps

\int \frac{e ^{2 x}}{5 + e ^{x}} d x

Apply u-substitution

= \int \frac{u - 5}{u} d u

Expand

\frac{u - 5}{u} : 1 - \frac{5}{u}

= \int 1 - \frac{5}{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

= \int 1 d u - \int \frac{5}{u} d u

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= u - 5 ln ∣ u ∣

Substitute back

u = 5 + e^{x}

= 5 + e^{x} - 5 ln ∣ 5 + e^{x} ∣

Add a constant to the solution

= 5 + e^{x} - 5 ln ∣ 5 + e^{x} ∣ + C

\frac{d}{d x} (3 x^{3} - 64 x)

x \to 1 + lim (tan (\frac{π x}{2}))

\int \frac{1}{2 u} d u

d er i v a t i v e (x^{2} - 6 x + 8)^{4}

\frac{d}{d x} (\frac{2 x + 1}{2 x - 1})

What is the integral of (e^{2x})/(5+e^x) ?
The integral of (e^{2x})/(5+e^x) is 5+e^x-5ln|5+e^x|+C