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

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

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

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

e^{x} + 3 - 3 ln ∣ e^{x} + 3 ∣ + C

Solution steps

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

Apply u-substitution

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

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

Expand

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

= \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

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

\int 1 d u = u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= u - 3 ln ∣ u ∣

Substitute back

u = e^{x} + 3

= e^{x} + 3 - 3 ln ∣ e^{x} + 3 ∣

Add a constant to the solution

= e^{x} + 3 - 3 ln ∣ e^{x} + 3 ∣ + C

\frac{\partial}{\partial x} (x ln (x + 7 y))

f (x) = (x^{2} + 8) (7 - x)

\int_{0}^{2} x^{7} e^{x^{4}} d x

\frac{d}{d x} ((2 x^{2} + 1)^{4})

x^{\frac{1}{4}}

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