Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
I edited your formatting. Please check, whether it is still correct. – k1next 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). – k1next 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
Okay, sorry my bad! – k1next Feb 18 '13 at 9:20

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 $y' = \sin(e^x)$.

share|cite|improve this answer

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

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.