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

Question

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

Answer 1

凑微分
因为xdx=(1/2)d(1+x^2)
∫ 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
∫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