Question

定积分∫x(e^sinx)|cosx|dx,上限是π,下线是0.急.

Answer 1

∫(0->π)x(e^sinx)|cosx|dx
=∫(0->π/2)x(e^sinx)(cosx)dx - ∫(π/2->π)x(e^sinx)(cosx)dx
let
y=π-x
dy = -dx
x=π/2 , y = π/2
x=π, y=0
-∫(π/2->π)x(e^sinx)(cosx)dx
=∫(0,π/2)(π-y)(e^siny)(cosy)dy
=∫(0,π/2)(π-x)(e^sinx)(cosx)dx

∫(0->π)x(e^sinx)|cosx|dx
=∫(0->π/2)x(e^sinx)(cosx)dx - ∫(π/2->π)x(e^sinx)(cosx)dx
=∫(0->π/2)x(e^sinx)(cosx)dx + ∫(0,π/2)(π-x)(e^sinx)(cosx)dx
=∫(0->π/2) π (e^sinx)(cosx)dx
= π [e^sinx](0->π/2)
=π(e-1)