x^1/2乘以cos(x^1/2)的不定积分

Question

如题所述

Answer 1

令√x = t、x = t^2、dx = 2t dt
∫ √xcos√x dx
= ∫ tcost * 2t dt
= 2∫ t^2 dsint
= 2t^2sint - 2∫ 2tsint dt
= 2t^2sint + 4∫ t dcost
= 2t^2sint + 4tcost - 4∫ cost dt
= 2t^2sint + 4tcost - 4sint + C
= 2xsin√x + 4√xcos√x - 4sin√x + C追问

看你做的真简单…为什么我都想到要把根号x变到后面的dx里去但就是不会令个t 代呢…烦死了

追答

没所谓、只要你不嫌麻烦的话、用凑微分法是可以的。
∫ √xcos√x dx
= ∫ (2√x√xcos√x)/(2√x) dx
= 2∫ xcos√x d√x
= 2∫ x d(sin√x)
= 2xsin√x - 2∫ sin√x dx
= 2xsin√x - 2∫ (2√xsin√x)/(2√x) dx
= 2xsin√x - 4∫ √xsin√x d√x
= 2xsin√x + 4∫ √x d(cos√x)
= 2xsin√x + 4√xcos√x - 4∫ cos√x d√x
= 2xsin√x + 4√xcos√x - 4sin√x + C

追问

好吧…膜拜之…圣诞快乐！