## Loser66 one year ago $$\int \dfrac{sin^2x}{7e^x}dx$$ Please, help

1. Loser66

@zepdrix

2. ganeshie8

use the identity $$\sin^2x = \frac{1-\cos (2x)}{2}$$, the integral becomes $\frac{1}{14}\int e^{-x}\, dx-\frac{1}{14}\int e^{-x}\cos(2x)\, dx$

3. Loser66

Yes, I did

4. ganeshie8

There are several ways to evaluate an integral of form $$\int e^{ax}\cos(bx)\, dx$$ there is really a very neat method if you're okay with complex numbers

5. Loser66

ah, you want to express it in term of Euler?

6. Loser66

I am ok with any method, please. show me. I can't get the answer as what wolfram does.

7. Loser66

8. ganeshie8

Yes : $\large e^{-x}\cos(2x) = \mathcal{R} (e^{-x+i2x})$

9. ganeshie8

we're done, integrating $$e^{-x+i2x}$$ is a piece of cake

10. ganeshie8

\begin{align} \int e^{-x}\cos(2x) \,dx &= \mathcal{R} \int e^{-x+i2x}\,dx\\~\\ &= \mathcal{R}~ \dfrac{e^{-x+i2x}}{-1+2i}\\~\\ \end{align} just get the real part of that expression and yeah don't forget the integration constant..

11. Loser66

Got you, thank you. Much appreciate. :)

12. ganeshie8

np :)