Evaluate Integral

Dec 2015
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Earth
Evaluate \(\displaystyle \int_{0}^{1} e^x \cos^{n}(x) dx \; , \: n \in \mathbb{N}\).
 
Jul 2008
5,231
51
Western Canada
Just curious, given the trigonometric term, why is your upper limit 1 instead of some multiple of π/4?
The antiderivative is listed in Dwight [576.9], for n=1, 2 or 3, and a general solution is given recursively involving a second integral in terms of n-2. If that's of any use to you, I can provide it.
 
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Jul 2008
5,231
51
Western Canada
$\displaystyle \int\ e^{a x} \cos^n (x)\, dx = \dfrac{e^{ax} \cos^{n-1}(x)}{a^2+n^2}(a \cos(x)+n\sin(x))+\dfrac{n(n-1)}{a^2+n^2}\displaystyle \int\ e^{a x} \cos^{n-2} (x)\, dx$
 
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