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

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

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

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

- \frac{1}{2} ln e^{2 x} + 2 + x + C

Solution steps

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

Take the constant out:

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

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

Simplify

\frac{e ^{x}}{e ^{3 x} + 2 e ^{x}} : \frac{1}{e ^{2 x} + 2}

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

Apply u-substitution

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

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

Take the constant out:

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

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

Complete the square

u (u - 2) : (u - 1)^{2} - 1

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{( u - 1 ) ^{2} - 1} d u

Apply u-substitution

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

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

Factor

v^{2} - 1 : - (- v^{2} + 1)

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= 2 \cdot \frac{1}{2} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

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

Simplify

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

= - \frac{1}{2} ln e^{2 x} + 2 + x

Add a constant to the solution

= - \frac{1}{2} ln e^{2 x} + 2 + x + C

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