Question

y=∫x(sinx)∧4dx 对x定积分,x从0到π

Answer 1

利用三角恒等式和分部积分
∫x(sinx)^4dx
= (3/8)∫xdx - (1/2)∫x*cos(2x)dx + (1/8)∫x*cos(4x)dx
= (3/16)x^2 - (1/2)*(1/2)[x*sin(2x)-∫sins(2x)dx] + (1/8)*(1/4)[x*sin(4x)-∫sin(4x)dx]
= (3/16)*x^2 - (1/4)x*sin(2x)-(1/8)*cos(2x) + (1/32)x*sin(4x)+(1/123)*cos(4x) + C
于是
y=∫x(sinx)∧4dx 对x定积分,x从0到π
= (3/16)*x^2 - (1/4)x*sin(2x)-(1/8)*cos(2x) + (1/32)x*sin(4x)+(1/123)*cos(4x) |_0^π
= (3/16)*π^2