Question

Evaluate:

`inte^x sinx  dx`

Answer 1

Let I = `inte^x sinx  dx`   ....(1)

Integrating by parts,

I = `e^x int sin x  dx - int(intsinx  dx d/dx e^x)dx`

= `e^x(-cosx) - int - cosx*e^xdx`

= `-e^x cosx + inte^x cosx  dx`

Again integrating by parts,

I = `-e^x cosx + e^x int cosx  dx - int(intcosx  dx d/dx e^x)dx`

I = `-e^x cosx + e^x sinx - int sinxe^x dx`

∴ I = `-e^x cosx + e^x sinx - I`  ....[From (1)]

∴ 2I = `-e^x(cosx - sinx) + c_1`

∴ I = `e^x/2(sinx - cosx) + c  ("where"  c = c_1/2)`