如图，求定积分，分部两次的过程如何写？

Question

如题所述

Answer 1

解如下图所示

追答

Answer 2

(2π/3)∫(0->π) e^(3x). sinx dx
=(2π/9)∫(0->π) sinx de^(3x)
=(2π/9)[e^(3x). sinx]|(0->π) -(2π/9)∫(0->π) e^(3x).cosx dx
=0-(2π/27)∫(0->π) cosx de^(3x)
=-(2π/27)[e^(3x).cosx]|(0->π) -(2π/27)∫(0->π) e^(3x).sinx dx
=(2π/27)(1+e^(3π).cosx ) -(2π/27)∫(0->π) e^(3x).sinx dx
(20π/27)∫(0->π) e^(3x). sinx dx =(2π/27)(1+e^(3π).cosx )
∫(0->π) e^(3x). sinx dx =(1/10)(1+e^(3π).cosx )
(2π/3)∫(0->π) e^(3x). sinx dx = (π/15)(1+e^(3π).cosx )