How do you integrate #int e^x sin sqrtx dx # using integration by parts?

2 Answers
May 11, 2018

I got overenthusiastic but I got stuck....I am not sure about it...I suspect it is either very complicated or not possible directly...


I got stuck...

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May 11, 2018

You do not.


You are not going to find a satisfactory answer to this integral. That is, the result cannot be represented by elementary functions. For reference, an acceptable result of this integral would be:

#int (e^xsin(sqrt(x)))dx = -(1/4)i(root4(e) sqrt(pi) " erf"(1/2 - isqrt(x)) - root4(e)sqrt(pi) " erf"(1/2 + isqrt(x)) + 2e^(x-isqrt(x))(-1+e^(2isqrt(x)))) + C#

where #"erf"(x) = 2/(sqrt(pi)) int_0^x e^(-t^2)dt# and #i# is the imaginary number.

You would not encounter this type of integral in a high school or college level calculus class. In fact, you would not see an integral of this type even while pursuing an undergraduate mathematics degree.