What is the integral of (3e^x)/(1-e^x) ?

The integral of (3e^x)/(1-e^x) is -3ln|1-e^x|+C

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integral of (3e^x)/(1-e^x)

\int \frac{3 e ^{x}}{1 - e ^{x}} d x

- 3 ln ∣ 1 - e^{x} ∣ + C

Solution steps

\int \frac{3 e ^{x}}{1 - e ^{x}} d x

Take the constant out:

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

= 3 \cdot \int \frac{e ^{x}}{1 - e ^{x}} d x

Apply u-substitution

: \int - \frac{1}{u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 (- ln ∣ u ∣)

Substitute back

u = 1 - e^{x}

= 3 (- ln ∣ 1 - e^{x} ∣)

Simplify

= - 3 ln ∣ 1 - e^{x} ∣

Add a constant to the solution

= - 3 ln ∣ 1 - e^{x} ∣ + C

x = y (y - 3), x = - y (y - 3)

\int - 20 t^{\frac{2}{3}} - 70 d t

\frac{3}{8 ( s + 1 )}

2 t d t + (y - 3 t) d y = 0

\frac{d}{d x} (2 - 5 x)

What is the integral of (3e^x)/(1-e^x) ?
The integral of (3e^x)/(1-e^x) is -3ln|1-e^x|+C