July 8th, 2019, 07:52 AM  #1 
Newbie Joined: Jul 2019 From: Romania Posts: 1 Thanks: 0  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^(n1))(cosx)—n(n1){I(n2)} 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. 
July 8th, 2019, 12:32 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,980 Thanks: 1573 
$\displaystyle I(n) = \int_{\pi}^\pi x^n \cos{x} \, dx$ 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 yaxis ... $\displaystyle I(2) = 2\int_0^\pi x^2 \cos{x} \, dx = 2\bigg[x^2\sin{x}+2x\cos{x}+2\sin{x}\bigg]_0^\pi = 2\bigg[(2\pi)  (0) \bigg] =4\pi$ 

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