What is the integral of (5e^{3x})/(1+e^{2x)} ?

The integral of (5e^{3x})/(1+e^{2x)} is 5(-arctan(e^x)+e^x)+C

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integral of (5e^{3x})/(1+e^{2x)}

\int \frac{5 e ^{3 x}}{1 + e ^{2 x}} d x

5 (- arctan (e^{x}) + e^{x}) + C

Solution steps

\int \frac{5 e ^{3 x}}{1 + e ^{2 x}} d x

Take the constant out:

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

= 5 \cdot \int \frac{e ^{3 x}}{1 + e ^{2 x}} d x

Apply u-substitution

: \int \frac{u - 1}{2 u} d u

= 5 \cdot \int \frac{u - 1}{2 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot \int \frac{u - 1}{u} d u

Apply u-substitution

: \int \frac{2 v ^{2}}{v ^{2} + 1} d v

= 5 \cdot \frac{1}{2} \cdot \int \frac{2 v ^{2}}{v ^{2} + 1} d v

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot 2 \cdot \int \frac{v ^{2}}{v ^{2} + 1} d v

\frac{v ^{2}}{v ^{2} + 1} = - \frac{1}{v ^{2} + 1} + 1

= 5 \cdot \frac{1}{2} \cdot 2 \cdot \int - \frac{1}{v ^{2} + 1} + 1 d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 \cdot \frac{1}{2} \cdot 2 (- \int \frac{1}{v ^{2} + 1} d v + \int 1 d v)

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

\int 1 d v = v

= 5 \cdot \frac{1}{2} \cdot 2 (- arctan (v) + v)

Substitute back

= 5 \cdot \frac{1}{2} \cdot 2 (- arctan (1 + e^{2 x} - 1) + 1 + e^{2 x} - 1)

Simplify

5 \cdot \frac{1}{2} \cdot 2 (- arctan (1 + e^{2 x} - 1) + 1 + e^{2 x} - 1) : 5 (- arctan (e^{x}) + e^{x})

= 5 (- arctan (e^{x}) + e^{x})

Add a constant to the solution

= 5 (- arctan (e^{x}) + e^{x}) + C

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What is the integral of (5e^{3x})/(1+e^{2x)} ?
The integral of (5e^{3x})/(1+e^{2x)} is 5(-arctan(e^x)+e^x)+C