7.1 Integration By Parts/24

${\begin{aligned}&\int _{0}^{\pi }\underbrace {x^{3}\cos(x)} _{\begin{aligned}u&=x^{3}\quad \quad dv=\cos(x)\\dv&=3x^{2}\quad \quad v=\sin(x)\end{aligned}}\,dx=x^{3}\sin(x)-\int _{0}^{\pi }\underbrace {3x^{2}\sin(x)} _{\begin{aligned}u&=3x^{2}\quad \quad dv=\sin(x)\\du&=6x\quad \quad v=-\cos(x)\end{aligned}}\,dx=x^{3}\sin(x)-[3x^{2}-\cos(x)-\int _{0}^{\pi }-6x\cos(x)\,dx]\\=&x^{3}\sin(x)-3x^{2}\cos(x)-\int _{0}^{\pi }6x\cos(x)\end{aligned}}$