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

The integral of e^2xcos(x) is e^2(xsin(x)+cos(x))+C

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integral of e^2xcos(x)

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

e^{2} (x sin (x) + cos (x)) + C

Solution steps

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

Take the constant out:

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

= e^{2} \cdot \int x cos (x) d x

Apply Integration By Parts

: x sin (x) - \int sin (x) d x

= e^{2} (x sin (x) - \int sin (x) d x)

\int sin (x) d x = - cos (x)

= e^{2} (x sin (x) - (- cos (x)))

Simplify

= e^{2} (x sin (x) + cos (x))

Add a constant to the solution

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

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What is the integral of e^2xcos(x) ?
The integral of e^2xcos(x) is e^2(xsin(x)+cos(x))+C