The question asks for the integral of $e^{mx^{3}}.$ How do I calculate the antiderivative of a function that has $e^{x^{3}}$ lets say. I want to make sure I know how to take the antiderivative of exponential functions.
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4Is it $e^m x^3$, $\left(e^{mx}\right)^3$ or $e^{mx^3}$? Exponentiation is not associative, hence e^x^3 is an ambiguous notation. – Jack D'Aurizio Feb 01 '17 at 00:36
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Its the third option you put – Chris Feb 01 '17 at 00:37
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Straight from Mathematica: $-\frac{x \Gamma \left(\frac{1}{3},-m x^3\right)}{3 \sqrt[3]{-m x^3}}$ – David G. Stork Feb 01 '17 at 00:40
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@DavidG.Stork Lol, you took the easy route to that answer. – Simply Beautiful Art Feb 01 '17 at 00:45
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Thanks for all the replies! I dont think my teacher meant to put this in the review. I haven't learned gamma yet idk what that is lol – Chris Feb 01 '17 at 00:47
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Repeated question: http://math.stackexchange.com/questions/270721/how-to-evaluate-the-integral-int-ex3dx – fsbmat Feb 01 '17 at 00:47
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@Chris You can do it with the exponential integral if that's what you meant. – Simply Beautiful Art Feb 01 '17 at 00:47
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Let $mt^3=-u,$
$$\int_0^xe^{mt^3}\ dt=-\int_0^{mx^3}\frac1{3\sqrt[3]{mu}}e^{-u}\ du=-\frac1{3\sqrt[3]{m}}\gamma\left(\frac23,mx^3\right)$$
where $\gamma(a,b)$ is the lower incomplete gamma function.
Simply Beautiful Art
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