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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

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 r} + 2

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

Simplify

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

= 3 ln e^{5 r} + 2

Add a constant to the solution

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

\int x^{2} \cdot ln (3 x) d x

y = - 3 cos (x), at x = \frac{3 π}{4}

\int (x + 1)^{3} d x

\frac{d}{d x} (\frac{2 x}{x ^{2} - 1})

x \to 0 - lim (- x)

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