Question

Show that $\int$ 𝑥 3 cos 𝑥 𝑑𝑥 𝜋 2 0 = 𝜋 3 8 − 3𝜋 + 6

Answer 1

$\int_0^{2 \pi} x^3\cos x dx= x^3\sin x\Biggr|_0^{2\pi}-\int_0^{2 \pi}3x^2 \sin x dx\\ =-3\left[-x^2 \cos x\Biggr|_0^{2 \pi}+2\int_0^{2 \pi}x \cos x dx\right]\\ =-3\left[-4 \pi^2+2\left(x \sin x\Biggr|_0^{2\pi}-\int_0^{2\pi}\sin x dx\right)\right]\\ =-3[-4\pi^2+2(\cos x\Biggr|_0^{2\pi})]\\ =-3[-4\pi^2+2]=12\pi^2-6$