求不定积分：∫ln(x^2+1)dx
问题描述：
求不定积分：∫ln(x^2+1)dx
答
∫ln(x^2+1)dx
=xln(x^2+1)-∫2x*x*1/(x²+1)dx
=xln(x^2+1)-2∫x²/(x²+1)dx
=xln(x^2+1)-2∫(x²+1-1)/(x²+1)dx
=xln(x^2+1)-2[∫1-1/(x²+1)dx]
=xln(x^2+1)+2[∫-1dx+∫1/(x²+1)dx]
=xln(x^2+1)-2x+2arctanx+C
答
用分步积分
∫ln(x^2+1)dx
=xln(x^2+1)-∫2x^2/(x^2+1)dx
=xln(x^2+1)-∫(2x^2+2-2)/(x^2+1)dx
=xln(x^2+1)-∫[2-2/(x^2+1)]dx
=xln(x^2+1)-2x+2arctanx+C