Question

Evaluate the integral using the properties of even and odd functions as an aid. (in comments)

Answer 1

\[\int\limits_{-2}^{2} x(x^2+1)^3 dx\]

Answer 2

Well \[(x ^{2}+1)^{3}=x ^{6}+3x ^{4}+3x ^{2}+1\]

Multiply everything in there by x and it looks like all odd powers of x to me.

Answer 3

@Batman_ProtectorofGotham

\[\int\limits_{-a}^{a}f(x)dx =2\int\limits_{0}^{a}f(x)dx\]if f(-x) = f(x). This implies that f(x) is even.
\[\int\limits_{-a}^{a}f(x)dx =0\]if f(-x) = -f(x) or f(x) = -f(-x). This implies that f(x) is odd.

Answer 4

thanks a bunch!

Answer 5

Welcome! I hope you're able to solve these problems now!

Answer 6

@Batman_ProtectorofGotham are you OK solving this now or do you still need help?