How can I integrate e^x sinx?

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Such an integral looks difficult to manage; clearly it is a case of integration by parts, but neither part of the product appears to reduce easily into a nicer form.

The key here is use the fact that sinx has a cyclic pattern when it is repatedley differentiated. Keeping our orignal integral on the left hand side, after two applications of integration by parts, where sinx is the term to be differentiated, we obtain a second intance of the integral on the right hand side of the equation. Moving this integral to the left hand side with our original integral and dividing the equation by two, we acquire the required result.

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