求x²arctanx的不定积分

求x²arctanx的不定积分
求x²arctanx的不定积分

求x²arctanx的不定积分
∫ x²arctanx dx
= ∫ arctanx d(x³/3)
= (1/3)x³arctanx - (1/3)∫ x³/(1+x²) dx,分部积分法
= (1/3)x³arctanx - (1/3)∫ x(x²+1-1)/(1+x²) dx
= (1/3)x³arctanx - (1/3)∫ xdx + (1/3)∫ x/(1+x²) dx
= (1/3)x³arctanx - x²/6 + (1/6)∫ 1/(1+x²) d(x²+1),凑微分法
= (1/3)x³arctanx - x²/6 + (1/6)ln|1+x²| + C

原式子
=(1/3)∫arctanxd(x^3)
=(1/3)x^3arctanx-(1/3)∫x^3dx/(1+x^2)
=(1/3)x^3arctanx-(1/3)∫xdx+(1/3)∫xdx/(1+x^2)
=(1/3)x^3arctanx-(1/6)x^2+(1/6)ln(1+x^2)+c.