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

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

Popular >

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

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

e^{x} + 5 - 5 ln ∣ e^{x} + 5 ∣ + C

Solution steps

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

Apply u-substitution

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

= \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 = e^{x} + 5

= e^{x} + 5 - 5 ln ∣ e^{x} + 5 ∣

Add a constant to the solution

= e^{x} + 5 - 5 ln ∣ e^{x} + 5 ∣ + C

\frac{\partial}{\partial x} (y (- 3 - x))

\int cos (x) 7 5 + sin (x) d x

\frac{\partial}{\partial x} (x y - x^{2} y)

\frac{\partial}{\partial x} (2 x z^{3} e^{y^{2}})

\int \frac{1}{1 + z ^{2}} d z

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