Question

Integrate the following w.r.t.x : e^2x sin x cos x

Answer 1

Let I = `int e^(2x)*sin x cos x*dx`

= `(1)/(2) int e(2x)*2sin x cos x dx`

= `(1)/(2) int e^(2x)*sin2x *dx`              ...(1)

= `(1)/(2)[e^(2x) int sin 2x*dx - int {d/dx (e^(2x)) int sin 2x*dx}*dx]`

= `(1)/(2)[e(2x) ((-cos2x)/2) - int e^(2x) xx 2 xx ((- cos2x)/2)*dx]`

= `-(1)/(4) e^(2x) cos 2x + 1/2 int e^(2x) cos 2x*dx`

= `-(1)/(4)e^(2x) cos2x + (1)/(2)[e^(2x) int cos 2x*dx - int {d/dx (e^(2x)) int cos 2x*dx }*dx]`

= `(1)/(4)e^(2x) cos 2x + 1/2 [e^(2x).(sin2x)/(2) - int e^(2x) xx 2 xx (sin2x)/(2)*dx]`

= `-(1)/(4) e^(2x) cos 2x + (1)/(4) e^(2x) sin 2x - (1)/(2) int e^(2x) sin2x*dx`

∴ I = `-(1)/(4) e^(2x) cos 2x + (1)/(4) e^(2x) sin 2x - "I"`   ..[By (1)]

∴ 2I = `-(1)/(4)e^(2x) cos 2x + 1/4e^(2x) sin2x`

∴ I = `e^(2x)/(8)(sin2x - cos2x) + c`.