What is the integral of (15e^{5x})/(e^{5x)+2} ?

The integral of (15e^{5x})/(e^{5x)+2} is 3ln|e^{5x}+2|+C

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integral of (15e^{5x})/(e^{5x)+2}

\int \frac{15 e ^{5 x}}{e ^{5 x} + 2} d x

3 ln e^{5 x} + 2 + C

Solution steps

\int \frac{15 e ^{5 x}}{e ^{5 x} + 2} d x

Take the constant out:

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

= 15 \cdot \int \frac{e ^{5 x}}{e ^{5 x} + 2} d x

Apply u-substitution

: \int \frac{1}{5 u} d u

= 15 \cdot \int \frac{1}{5 u} d u

Take the constant out:

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

= 15 \cdot \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 15 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = e^{5 x} + 2

= 15 \cdot \frac{1}{5} ln e^{5 x} + 2

Simplify

15 \cdot \frac{1}{5} ln e^{5 x} + 2 : 3 ln e^{5 x} + 2

= 3 ln e^{5 x} + 2

Add a constant to the solution

= 3 ln e^{5 x} + 2 + C

f (x) = - 4 x^{2} + 7 x - 14, at x = - 2

\frac{\partial}{\partial x} (x^{c o s (x)})

e^{18 x}

x \to \infty lim (x^{(\frac{1}{x ^{2} - 1})})

\int (x^{2} - 1)^{4} d x

What is the integral of (15e^{5x})/(e^{5x)+2} ?
The integral of (15e^{5x})/(e^{5x)+2} is 3ln|e^{5x}+2|+C