x*根号|x|的定积分,3到-2你题目看错了

Question

x*根号|x|的定积分,3到-2
你题目看错了

Answer 1

不清楚本题的积分区间是3→-2,还是-2→3,
现分两种情况
∫x√|x|dx (x:3→-2)
= -∫x√|x|dx (x:-2→3)
= -∫x√|x|dx (x:-2→0) - ∫x√|x|dx (x:→3)
令 -x = u, dx = - du, y:2→0
∴原积分
= -∫(-u)(√u)(-du) (u:2→0) - ∫x√xdx (x:→3)
= -∫u^(3/2)du (u:2→0) - ∫x^(3/2)dx (x:→3)
= ∫u^(3/2)du (u:0→2) - ∫x^(3/2)dx (x:→3)
= (2/5)u^(5/2)(u:0→2) - (2/5)x^(5/2)(x:0→3)
= (2/5)4√2 - (2/5)9√3
= (2/5)(4√2 - 9√3)
≈ -3.97264
∫x√|x|dx (x:-2→3)
=-∫x√|x|dx (x:-2→3)
= (2/5)(9√3 - 4√2)
≈ 3.97264