计算下列积分!SOS∫0→5 x^3/x^2+1dx ∫0→1 xdx/x^2+1

凑微分 因为xdx=(1/2)d(1+x^2) 1.∫ x^3/x^2+1dx =(1/2)∫x^2/x^2+1d(1+x^2) 令t=1+x^2 =(1/2)∫(t-1)dt/t =(1/2)[∫dt-∫dt/t] =(1/2)(t-ln|t|)+C =(1/2)(1+x^2-ln(1+x^2))+C 代入x=5,0相减得 (1/2)(25-ln(26)) 类似 2.∫xdx/x^2+1 =(1/2)∫d(1+x^2)/(1+x^2) =(1/2)ln(1+x^2)+C 代入x=1,x=0相减得 (1/2)ln 2