求不定积分∫ x arcsin(x/2) dx

题目

求不定积分∫ x arcsin(x/2) dx

答案

用分部积分法+三角换元法：
∫ xarcsin(x / 2) dx
令y = x / 2
= 4∫ yarcsiny dy
= 2∫ arcsiny d(y²)
= 2y²arcsiny - 2∫ y²d(arcsiny)
= 2y²arcsiny -2∫ y² / √(1 - y²) dy
令y = sinA,dy = cosA dA
√(1 - y²) = √(1 - sin²A) = cosA
= 2y²arcsiny -2∫ sin²A / cosA * cosA dA
= 2y²arcsiny -∫ (1 - cos2A) dA
= 2y²arcsiny -(A - sin2A / 2) + C
= 2y²arcsiny - A + sinAcosA + C
= 2y²arcsiny - arcsiny + y√(1 - y²) + C
= (2 * x² / 4 - 1)arcsin(x / 2) + (x / 2)√[1 - (x / 2)²] + C
= (1 / 2)(x² - 2)arcsin(x / 2) + (x / 4)√(4 - x²) + C