∫根号x*sin根号xdx怎么算急谢咯

Question

∫根号x*sin根号xdx 怎么算 急 谢咯

Answer 1

∫ √xsin√x dx
令y = √x,dy = 1/2√x dx => dx = 2y dy
= ∫ ysiny • 2y dy = 2∫ y²siny dy = -2∫ y² dcosy
= -2y²cosy + 2∫ cosy dy²,分部积分法
= -2y²cosy + 4∫ ycosy dy
= -2y²cosy + 4∫ y dsiny
= -2y²cosy + 4ysiny - 4∫ siny dy,分部积分法
= -2y²cosy + 4ysiny + 4cosy + C
= -2xcos√x + 4√xsin√x + 4cos√x + C,回代x = y²