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July 8th, 2019, 07:52 AM   #1
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Fully defined integrals

Hello!
So I've got this integral:
https://prnt.sc/oc6zok
And I managed to get to this result : I(n)=x^(n)(sinx)+n(x^(n-1))(cosx)—n(n-1){I(n-2)}

However, since the integral is defined on the interval (-pi,pi) I don't know how to continue.

Excuse my English for I am not a native English speaker and I'm not used to utilize mathematical terms in this language.I basically don't know how to proceed further.I know that somehow I've got to give x the values of -pi and pi but I don't know how to get to that point.
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July 8th, 2019, 12:32 PM   #2
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$\displaystyle I(n) = \int_{-\pi}^\pi x^n \cos{x} \, dx$ for $n \in \{2,3 \}$


$I(3) = 0$ because the integrand, $x^3\cos{x}$, is an odd function symmetric to the origin.


For $n=2$, the integrand, $x^2\cos{x}$, is an even function. Even functions have symmetry across the y-axis ...

$\displaystyle I(2) = 2\int_0^\pi x^2 \cos{x} \, dx = 2\bigg[-x^2\sin{x}+2x\cos{x}+2\sin{x}\bigg]_0^\pi = 2\bigg[(-2\pi) - (0) \bigg] =-4\pi$
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