∫ x*cscx^2*cotx^2 dx

∫ x*csc^2x*cot^2x dx
=(1/3)*∫ x*(3*csc^2x*cot^2x) dx
=(1/3)*∫ x d(cot^3x) ……凑微分法
=(1/3)*x*cot^3x - (1/3)*∫ cot^3x dx ……分部积分法
=(1/3)*x*cot^3x - (1/3)*∫ cos^3x/sin^3x dx
=(1/3)*x*cot^3x - (1/3)*∫ cos^2x/sin^3x d(sinx) ……凑微分法
=(1/3)*x*cot^3x - (1/3)*∫ (1-sin^2x)/sin^3x d(sinx)
=(1/3)*x*cot^3x - (1/3)*∫ 1/sin^3x d(sinx) + (1/3)*∫ 1/sinx d(sinx)
=(1/3)*x*cot^3x + (1/6)*(1/sin^2x) + (1/3)*ln |sinx| +C
有不懂欢迎追问