∫﹛﹙x－1﹚／﹙x2＋x＋1﹚﹜dx怎么做

题目

∫﹛﹙x－1﹚／﹙x2＋x＋1﹚﹜dx怎么做

答案

∫ (x - 1)/(x² + x + 1) dx
= ∫ {(1/2)[(2x + 1) - 1] - 1}/(x² + x + 1) dx
= (1/2)∫ (2x + 1)/(x² + x + 1) dx - (3/2)∫ dx/(x² + x + 1)
= (1/2)∫ d(x² + x + 1)/(x² + x + 1) - (3/2)∫ d(x + 1/2)/[(x + 1/2)² + 3/4]
= (1/2)ln|x² + x + 1| - (3/2)(2/√3)arctan[(x + 1/2)(2/√3)] + C
= (1/2)ln|x² + x + 1| - √3arctan[(2x + 1)/√3] + C