What is the integral of (4e^x)/(7-3e^x) ?

The integral of (4e^x)/(7-3e^x) is -4/3 ln|7-3e^x|+C

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integral of (4e^x)/(7-3e^x)

\int \frac{4 e ^{x}}{7 - 3 e ^{x}} d x

- \frac{4}{3} ln ∣ 7 - 3 e^{x} ∣ + C

Solution steps

\int \frac{4 e ^{x}}{7 - 3 e ^{x}} d x

Take the constant out:

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

= 4 \cdot \int \frac{e ^{x}}{7 - 3 e ^{x}} d x

Apply u-substitution

= 4 \cdot \int - \frac{1}{3 u} d u

Take the constant out:

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

= 4 (- \frac{1}{3} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 (- \frac{1}{3} ln ∣ u ∣)

Substitute back

u = 7 - 3 e^{x}

= 4 (- \frac{1}{3} ln ∣ 7 - 3 e^{x} ∣)

Simplify

4 (- \frac{1}{3} ln ∣ 7 - 3 e^{x} ∣) : - \frac{4}{3} ln ∣ 7 - 3 e^{x} ∣

= - \frac{4}{3} ln ∣ 7 - 3 e^{x} ∣

Add a constant to the solution

= - \frac{4}{3} ln ∣ 7 - 3 e^{x} ∣ + C

What is the integral of (4e^x)/(7-3e^x) ?
The integral of (4e^x)/(7-3e^x) is -4/3 ln|7-3e^x|+C