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Question
Integrate the following w.r.t.x : e2x sin x cos x
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Solution
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`.
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