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 Calculus Calculus Math Forum

 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^(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. July 8th, 2019, 12:32 PM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 3,016 Thanks: 1600 $\displaystyle I(n) = \int_{-\pi}^\pi x^n \cos{x} \, dx$ 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 ... $\displaystyle I(2) = 2\int_0^\pi x^2 \cos{x} \, dx = 2\bigg[-x^2\sin{x}+2x\cos{x}+2\sin{x}\bigg]_0^\pi = 2\bigg[(-2\pi) - (0) \bigg] =-4\pi$ Thanks from topsquark Tags defined, fully, integrals Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post dennisdixon New Users 8 March 3rd, 2013 07:09 AM durky Abstract Algebra 6 March 28th, 2012 01:29 PM probiner Algebra 4 February 17th, 2012 11:49 AM jonas Algebra 1 November 12th, 2009 10:55 PM

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