What is the integral of e^xcos(e^x)+(2x+2)/(x^2+2x) ?

The integral of e^xcos(e^x)+(2x+2)/(x^2+2x) is sin(e^x)+ln|x+2|+ln|x|+C

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integral of e^xcos(e^x)+(2x+2)/(x^2+2x)

\int e^{x} cos (e^{x}) + \frac{2 x + 2}{x ^{2} + 2 x} d x

sin (e^{x}) + ln ∣ x + 2 ∣ + ln ∣ x ∣ + C

Solution steps

\int e^{x} cos (e^{x}) + \frac{2 x + 2}{x ^{2} + 2 x} 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 e^{x} cos (e^{x}) d x + \int \frac{2 x + 2}{x ^{2} + 2 x} d x

\int e^{x} cos (e^{x}) d x = sin (e^{x})

\int \frac{2 x + 2}{x ^{2} + 2 x} d x = ln ∣ x + 2 ∣ + ln ∣ x ∣

= sin (e^{x}) + ln ∣ x + 2 ∣ + ln ∣ x ∣

Add a constant to the solution

= sin (e^{x}) + ln ∣ x + 2 ∣ + ln ∣ x ∣ + C

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What is the integral of e^xcos(e^x)+(2x+2)/(x^2+2x) ?
The integral of e^xcos(e^x)+(2x+2)/(x^2+2x) is sin(e^x)+ln|x+2|+ln|x|+C