Is it possible to integrate $e^{x}\ \sinh(x)$ using only integration by parts?

Question

Is it possible to integrate $e^{x}\ \sinh(x)$ using only integration by parts?

If so, how would it be done?

I read the following Stack Overflow article, Integrate $\int e^x \cosh(x) \: dx$ by parts, which contains an arithmetic mistake.

Any help is greatly appreciated.

Answer 1

$\int e^x \sinh x = e^x \sinh x-\int e^x \cosh x=e^x \sinh x - \int e^x (\sinh x + e^{-x})=e^x \sinh x - \int e^x \sinh x - \int 1 = e^x \sinh x - x - \int e^x \sinh x $

Gives: $\int e^x \sinh x = \frac{e^x \sinh x-x}{2}$

Edit: I am using $\cosh x = \sinh x +e^{-x}$.

Answer 2

$e^x \sinh(x) = e^x \frac{e^x - e^{-x}}{2} = \frac{1}{2} \left( e^{2x} -1 \right)$. Why would you want to use integration by parts?

Answer 3

Sure. Set $\mathrm{e}^x = \cos(-\mathrm{i}x) + \mathrm{i} \sin( -\mathrm{i} x)$ and then use the usual method for trigonometric functions.

Although Jon Warneke's answer is a much better way to attack this integral.