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Question
Integrate the function in e2x sin x.
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Solution
Let `I = inte^(2x) sinx dx`
`= e^(2x) int sin x dx - int [d/dx (e^(2x))* int sin x dx] dx`
`= e^(2x) (- cos x) - int 2e^(2x) (- cos x) dx + C_1`
`= -e^(2x) cos x + 2 int e^(2x) cos x dx + C_1`
`= -e^(2x) cos x + 2` `...[e^(2x) int cos x dx - int (d/dx (e^(2x))* int cos xdx) dx] + C_1`
`= -e^(2x) cos x + 2e^(2x) sin x - 4 int e^(2x) sin x dx + C_1 + C_2`
`= e^(2x) (2 sin x - cos x) - 4I + C_1 + C_2`
∵ `5I = e^(2x) (2 sinx - cos x) + C_1 + C_2`
⇒ `I = (e^(2x))/5 (2 sin x - cos x) + C`
where C = C1 + C2
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