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

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ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2use the identity \(\sin^2x = \frac{1\cos (2x)}{2}\), the integral becomes \[\frac{1}{14}\int e^{x}\, dx\frac{1}{14}\int e^{x}\cos(2x)\, dx\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2There 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

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1ah, you want to express it in term of Euler?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1I am ok with any method, please. show me. I can't get the answer as what wolfram does.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2Yes : \[\large e^{x}\cos(2x) = \mathcal{R} (e^{x+i2x})\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2we're done, integrating \(e^{x+i2x}\) is a piece of cake

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2\[\begin{align} \int e^{x}\cos(2x) \,dx &= \mathcal{R} \int e^{x+i2x}\,dx\\~\\ &= \mathcal{R}~ \dfrac{e^{x+i2x}}{1+2i}\\~\\ \end{align}\] just get the real part of that expression and yeah don't forget the integration constant..

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1Got you, thank you. Much appreciate. :)
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