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Consider \begin{align} y = \int \sin(e^ x) \ dx \end{align} How do I integrate that, I know the result is $y'=\sin(e^x)$ but why?

So, I can't do it straight away, so I've got to use either substitution or integration by parts, as it is a composition I would say substitution but I don't see any relation between $\cos$ and $e^x$ ...any detailed help about how to proceed? thanks!

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I edited your formatting. Please check, whether it is still correct. –  sonystarmap Feb 18 '13 at 9:10
    
With the formatting change I'm now unsure, do you want to integrate that or differentiate $y$? –  Jim Feb 18 '13 at 9:15
    
@Jim: All I edited was replacing y = ∫sin(e^x)dx with the Latex formula and putting dollar signs around y'=sin(e^x). –  sonystarmap Feb 18 '13 at 9:19
    
Yea, I just meant that the new formatting made me read the question more carefully. –  Jim Feb 18 '13 at 9:19
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Okay, sorry my bad! –  sonystarmap Feb 18 '13 at 9:20

2 Answers 2

I deleted my old answer for the following reason: While you can do a substitution the resulting integral can't be expressed in terms of elementary functions. Trying to evaluate that integral symbolically is essentially impossible.

Given that your book says the answer is $y' = \sin(e^x)$ I think the problem is to take the derivative of $y$, not to evaluate that integral.

To take the derivative of $y$ you use the fact that without bounds the symbols $\int f(x)dx$ just mean "the function whose derivative is $f(x)$". Hence if we take the derivative of the function whose derivative is $f(x)$ we get $f(x)$. Hence $y' = \sin(e^x)$.

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$I=\int \sin(e^x) \, dx$

Let $t=e^x$

$$I=\int \frac {\sin(t)}{t} \, dt= \operatorname{Si}(e^x)+C$$

As far as I know, there is no solution in terms of elementary functions.

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