(x^2)*sin^2(πx)积分怎么算前面是X平后面是正弦函数平π是圆周率常数如题急等

Question

(x^2)*sin^2(πx)积分怎么算 前面是X平 后面是正弦函数平 π是圆周率常数
如题 急等

Answer 1

先降幂,再分部积分.
∫x^2*(sinπx)^2dx = (1/2)∫x^2[1-cos2πx]dx
=x^3/6-[1/(4π)]∫x^2dsin2πx
=x^3/6-[1/(4π)]x^2sin2πx+[1/(2π)]∫xsin2πxdx
=x^3/6-[1/(4π)]x^2sin2πx-[1/(2π)^2]∫xdcos2πx
=x^3/6-[1/(4π)]x^2sin2πx-[1/(2π)^2]xcos2πx+[1/(2π)^2]∫cos2πxdx
=x^3/6-[1/(4π)]x^2sin2πx-[1/(2π)^2]xcos2πx+[1/(2π)^3]sin2πx+C.