What is the integral of (5e^x)/(e^{2x)+4e^x+4} ?

The integral of (5e^x)/(e^{2x)+4e^x+4} is -5/(e^x+2)+C

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integral of (5e^x)/(e^{2x)+4e^x+4}

\int \frac{5 e ^{x}}{e ^{2 x} + 4 e ^{x} + 4} d x

- \frac{5}{e ^{x} + 2} + C

Solution steps

\int \frac{5 e ^{x}}{e ^{2 x} + 4 e ^{x} + 4} 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 ^{x}}{e ^{2 x} + 4 e ^{x} + 4} d x

Apply u-substitution

: \int \frac{1}{u ^{2} + 4 u + 4} d u

= 5 \cdot \int \frac{1}{u ^{2} + 4 u + 4} d u

Complete the square

u^{2} + 4 u + 4 : (u + 2)^{2}

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

Apply u-substitution

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

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

Apply the Power Rule

: - \frac{1}{v}

= 5 (- \frac{1}{v})

Substitute back

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

Simplify

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

= - \frac{5}{e ^{x} + 2}

Add a constant to the solution

= - \frac{5}{e ^{x} + 2} + C

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What is the integral of (5e^x)/(e^{2x)+4e^x+4} ?
The integral of (5e^x)/(e^{2x)+4e^x+4} is -5/(e^x+2)+C