Integration of Product of e, with variable exponent, and a trigonometric function

I have been asked to find the definite integral, as described above, of a product of e, with a variable exponent, and a trigonometric function; an example would be the integral of e^x.cos(x). I have tried to use the integration by parts, or integration by reverse product rule differentiation (integral of udv/dx = uv - integral of vdu/dx), but have almost immediately realised that because both the differentiation and integration either of e with a variable exponent, or a trigonomnetric function, does not gradually cancel out the variables but increases the complexity of the function, the integral will simply increase to an infinite length. I therefore assumed this technique to be incorrect, and I want to ask if anyone can help me with the proper technique? Any help would be much appreciated!

Re: Integration of Product of e, with variable exponent, and a trigonometric function

Re: Integration of Product of e, with variable exponent, and a trigonometric function

Thanks so much Soroban! I would never have considered that if the integral is expanded out to produce extra terms, the second integral of v(du/dx) in the supposedly infinite chain - if you follow me - would be a cosine and therefore the original integral or a factor of it, which can then be substituted; however, I would also never have used v = - cos x for that second integration of v(du/dx), that was true genius. Thanks again for taking the time to help me out, I really appreciate it, you are a true legend made of win. Peace.

Re: Integration of Product of e, with variable exponent, and a trigonometric function

Scratch that, only just remembered that the integral of sin x is - cos x . Derp derp derp, that tends to happen at 1 AM. Thanks again!