∫ln(x^2+1)dx,怎么算

来源:学生作业帮助网 编辑:作业帮 时间:2024/11/16 04:27:07

∫ln(x^2+1)dx,怎么算
∫ln(x^2+1)dx,怎么算

∫ln(x^2+1)dx,怎么算
分部积分
∫ln(x^2+1)dx = ∫x d ln(x^2+1) = xln(x^2+1) - ∫x d ln(x^2+1)
= xln(x^2+1) - 2∫(x^2/x^2+1)dx
= xln(x^2+1) - 2∫(x^2+1-1)/(x^2+1)dx
= xln(x^2+1) - 2[∫(x^2+1)/(x^2+1)dx -∫(1/x^2+1)dx]
= xln(x^2+1) - 2[∫dx -∫(1/x^2+1)dx]
= xln(x^2+1) - 2[x - arctanx]+C

∫ln(x^2+1)dx=∫xln(x^2+1)/xdx=∫ln(x^2+1)/xdx^2=∫xd1/(x^2+1)=x/(x^2+1)+∫1/(x^2+1)dx=arctanx

分部积分法:
=xln(x^2+1)-∫xdln(x^2+1)
=xln(x^2+1)-2∫x^2/(x^2+1)dx
=xln(x^2+1)-2∫(1-1/(x^2+1))
=xln(x^2+1)-(2x-arctanx)+C