Question

高数求定积分

Answer 1

let
y=x-2
dy = dx
x=1, y=-1
x=3, y=1

∫(1->3) f(x-2) dx
=∫(-1->1) f(y) dy
=∫(-1->0) (1+y^2) dy + ∫(0->1) e^(-y) dy
=[y +(1/3)y^3]|(-1->0) - [e^(-x)]|(0->1)
=4/3 -( e^(-1) - 1)
=7/3 - e^(-1)

Answer 2

∫[1,3]f(x-2)dx
=∫[1,2](1+(x-2)^2)dx + ∫[2,3]e^(-x-2)dx
=∫[1,2](x^2-4x+5)dx - ∫[2,3]e^(-x-2)d(-x-2)
=(1/3x^3-2x^2+5x)[1,2]-e^(-x-2)[2,3]
=(8/3-8+10)-(1/3-2+5)-e^(-5)+e^(-4)