What is the integral of (e^x)/(e^{2x)+9e^x-22} ?

The integral of (e^x)/(e^{2x)+9e^x-22} is 1/13 ln|e^x-2|-1/13 ln|e^x+11|+C

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integral of (e^x)/(e^{2x)+9e^x-22}

\int \frac{e ^{x}}{e ^{2 x} + 9 e ^{x} - 22} d x

\frac{1}{13} ln ∣ e^{x} - 2 ∣ - \frac{1}{13} ln ∣ e^{x} + 11 ∣ + C

Solution steps

\int \frac{e ^{x}}{e ^{2 x} + 9 e ^{x} - 22} d x

Take the partial fraction of

\frac{e ^{x}}{e ^{2 x} + 9 e ^{x} - 22} : \frac{2}{13 ( e ^{x} - 2 )} + \frac{11}{13 ( e ^{x} + 11 )}

= \int \frac{2}{13 ( e ^{x} - 2 )} + \frac{11}{13 ( e ^{x} + 11 )} d x

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{2}{13 ( e ^{x} - 2 )} d x + \int \frac{11}{13 ( e ^{x} + 11 )} d x

\int \frac{2}{13 ( e ^{x} - 2 )} d x = - \frac{1}{13} (x - ln ∣ e^{x} - 2 ∣)

\int \frac{11}{13 ( e ^{x} + 11 )} d x = \frac{1}{13} (x - ln ∣ e^{x} + 11 ∣)

= - \frac{1}{13} (x - ln ∣ e^{x} - 2 ∣) + \frac{1}{13} (x - ln ∣ e^{x} + 11 ∣)

Simplify

- \frac{1}{13} (x - ln ∣ e^{x} - 2 ∣) + \frac{1}{13} (x - ln ∣ e^{x} + 11 ∣) : \frac{1}{13} ln ∣ e^{x} - 2 ∣ - \frac{1}{13} ln ∣ e^{x} + 11 ∣

= \frac{1}{13} ln ∣ e^{x} - 2 ∣ - \frac{1}{13} ln ∣ e^{x} + 11 ∣

Add a constant to the solution

= \frac{1}{13} ln ∣ e^{x} - 2 ∣ - \frac{1}{13} ln ∣ e^{x} + 11 ∣ + C

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What is the integral of (e^x)/(e^{2x)+9e^x-22} ?
The integral of (e^x)/(e^{2x)+9e^x-22} is 1/13 ln|e^x-2|-1/13 ln|e^x+11|+C