Find ∫_(-π)^π(x³cos x + x²)dx. The function x³cos x is odd, because cos x is even and x³ is odd…

Step 1. Odd part. $f(x) = x^3\cos x$ has $f(-x) = -x^3\cos x = -f(x)$. Over $[-\pi, \pi]$ its integral is $0$.

Step 2. Even part. $\displaystyle\int_{-\pi}^{\pi}x^2\,dx = 2\int_0^\pi x^2\,dx = 2 \times \dfrac{\pi^3}{3} = \dfrac{2\pi^3}{3}$.

Step 3. Evaluate. $\pi^3 = 31.0063$, so $\dfrac{2 \times 31.0063}{3} = 20.6709$.