What is the integral of (e^x)/(e^{2x)+5e^x+6} ?

The integral of (e^x)/(e^{2x)+5e^x+6} is ln|e^x+2|-ln|e^x+3|+C

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integral of (e^x)/(e^{2x)+5e^x+6}

\int \frac{e ^{x}}{e ^{2 x} + 5 e ^{x} + 6} d x

ln ∣ e^{x} + 2 ∣ - ln ∣ e^{x} + 3 ∣ + C

Solution steps

\int \frac{e ^{x}}{e ^{2 x} + 5 e ^{x} + 6} d x

Apply u-substitution

= \int \frac{1}{u ^{2} + 5 u + 6} d u

Take the partial fraction of

\frac{1}{u ^{2} + 5 u + 6} : \frac{1}{u + 2} - \frac{1}{u + 3}

= \int \frac{1}{u + 2} - \frac{1}{u + 3} d u

Apply the Sum Rule:

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

= \int \frac{1}{u + 2} d u - \int \frac{1}{u + 3} d u

\int \frac{1}{u + 2} d u = ln ∣ u + 2 ∣

\int \frac{1}{u + 3} d u = ln ∣ u + 3 ∣

= ln ∣ u + 2 ∣ - ln ∣ u + 3 ∣

Substitute back

u = e^{x}

= ln ∣ e^{x} + 2 ∣ - ln ∣ e^{x} + 3 ∣

Add a constant to the solution

= ln ∣ e^{x} + 2 ∣ - ln ∣ e^{x} + 3 ∣ + C

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What is the integral of (e^x)/(e^{2x)+5e^x+6} ?
The integral of (e^x)/(e^{2x)+5e^x+6} is ln|e^x+2|-ln|e^x+3|+C