Antiderivative of $3\cos(x^3)\sin{x}$

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I am practicing for an exam by randomly picking an expression to anti-differentiate when I came up with this one: $3\cos(x^3)\sin{x}$ . How would I go about tackling this? I tried using integration by parts: $$\int3\cos(x^3)\sin{x}dx = 3\cos(x^3)(-\cos{x})-\int\sin(x)*-9x^2\sin(x^3) dx$$

This just ends up leaving the integral on the RHS even more complicated, in my opinion. I would appreciate it if somebody could show me how to solve this antiderivative.

Thanks in advance.

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1 Answer

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It might be a bad idea to just randomly come up with integrals and attempt to integrate them. There are a lot of functions with no anti-derivative expressible in elementary functions. In particular, $e^{x^2}$ is an example. According to wolframalpha, this integral is also not expressible in terms of elementary functions.

A lot of the problems you see in your textbook and from class are chosen to be solvable using the techniques you used. So it would be a much better idea to do extra problems in your textbook rather than coming up with random integrals, as the textbook problems will have a solution, while your problem may not.

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