xe的x次方除以根号下e的爱克死次方剪二的不定积分

题目

xe的x次方除以根号下e的爱克死次方剪二的不定积分

答案

∫ xe^x/√(e^x - 2) dx
-->令v = √(e^x - 2),v² = e^x - 2,2v dv = e^x dx-->
= ∫ ln(v² + 2) · e^x/v · 2v/(e^x) dv
= 2∫ ln(v² + 2) dv
= 2vln(v² + 2) - 2∫ v dln(v² + 2),分部积分法
= 2x√(e^x - 2) - 4∫ v²/(v² + 2) dv
= 2x√(e^x - 2) - 4∫ [(v² + 2) - 2]/(v² + 2) dv
= 2x√(e^x - 2) - 4∫ [1 - 2/(v² + 2)] dv
= 2x√(e^x - 2) - 4v + 8 · 1/√2 · arctan(v/√2) + C
= 2x√(e^x - 2) - 4√(e^x - 2) + 8 · (1/√2)(√2/√2) · arctan[√(e^x - 2)/√2] + C
= 2(x - 2)√(e^x - 2) + (4√2)arctan[√(e^x - 2)/√2] + C