题目

arctanx/x²(1+x²)的原函数

提问时间：2020-11-30

答案

arctan(x)/(x²(1+x²)) = arctan(x)/x²-arctan(x)/(1+x²),两项分别计算.
前者分部积分化为 -arctan(x)/x+∫1/(x(1+x²)) dx
= -arctan(x)/x+1/2·∫(1/x²-1/(1+x²))d(x²)
= -arctan(x)/x+1/2·∫1/x² d(x²)-1/2·∫1/(1+x²) d(1+x²)
= -arctan(x)/x+1/2·ln(x²)-1/2·ln(1+x²).
后者∫ arctan(x)/(1+x²) dx = ∫ arctan(x) d(arctan(x)) = arctan(x)²/2.