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Question
Find: `int e^x.sin2xdx`
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Solution
Let I = `int e^xsin2xdx`
Applying integration by parts
= `e^x int sin 2xdx - int [d/(dx) (e^x) int sin 2xdx]dx`
= `e^x((-cos2x)/2) + 1/2 int e^x cos 2xdx`
= `1/2(-e^x cos2x) + 1/2[e^x int cos 2xdx - int (d/(dx) (e^x) int cos2xdx)dx]`
= `1/2 (-e^x cos2x) + 1/2[(e^xsin2x)/2 - 1/2 int e^x sin 2xdx]`
= `1/2 (-e^x cos 2x) + 1/4 (e^x sin 2x) - 1/4 int e^x sin 2xdx + K`
∴ 4I = `-2e^x cos2x + e^xsin2x - I + K`
or 5I = `-2e^x cos2x + e^xsin2x + K`
I = `1/5(e^xsin2x - 2e^xcos2x) + K/5`
or I = `1/5(e^xsin2x - 2e^xcos2x) + c`
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