What is the integral of-(12e^{3x})/(e^{3x)+4} ?

The integral of-(12e^{3x})/(e^{3x)+4} is -4ln|e^{3x}+4|+C

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integral of-(12e^{3x})/(e^{3x)+4}

\int - \frac{12 e ^{3 x}}{e ^{3 x} + 4} d x

- 4 ln e^{3 x} + 4 + C

Solution steps

\int - \frac{12 e ^{3 x}}{e ^{3 x} + 4} d x

Take the constant out:

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

= - 12 \cdot \int \frac{e ^{3 x}}{e ^{3 x} + 4} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

= - 12 \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

= - 12 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 12 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = e^{3 x} + 4

= - 12 \cdot \frac{1}{3} ln e^{3 x} + 4

Simplify

- 12 \cdot \frac{1}{3} ln e^{3 x} + 4 : - 4 ln e^{3 x} + 4

= - 4 ln e^{3 x} + 4

Add a constant to the solution

= - 4 ln e^{3 x} + 4 + C

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What is the integral of-(12e^{3x})/(e^{3x)+4} ?
The integral of-(12e^{3x})/(e^{3x)+4} is -4ln|e^{3x}+4|+C