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Here is the problem: integrate e to the power 2xcos(x).

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Find the indefinite integral (antiderivative) of e to the power 2x cos(x) with respect to x.

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Start with the integral.

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Integrate by parts twice. Integrating by parts twice returns the original integral I, so I equals e^ ax (a cos bx plus b sin bx)/(a squared plus b squared) with a equals 2, b equals 1.

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Exponential times trigonometric (by parts twice).

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Simplify and add the constant of integration.

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So the answer is e to the power 2x times (2cos(x) plus sin(x)) / 5 plus C.

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An antiderivative F satisfies F prime of x equals f(x). Because the derivative of any constant is 0, every antiderivative differs by a constant C.

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Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.