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Question
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
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Solution
Let I = `int e^(2x).sin3x`
I = ` int sin 3x . e^(2x) dx`
I = `sin3x . int e^(2x) dx - int[d/dx (sin3x) int e^(2x)dx]dx`
I = `sin3x . e^(2x)/2 - int 3cos3x . e^(2x)/2 dx`
I = `1/2 sin3x.e^(2x) - 3/2 int cos3x . e^(2x)dx`
I = `1/2sin3x.e^(2x) - 3/2 intcos3x inte^(2x)dx - int [d/dx cos3x . int e^(2x)dx]dx`
I = `1/2 sin3x . e^(2x) - 3/2 cos3x . e^(2x)/2 + 3/2 int -sin3x . x3 . e^(2x)/2 dx`
I = `1/2 sin3x . e^(2x) - 3/4 cos3x . e^(2x) - 9/4 [int sin3x . e^(2x) dx]`
I = `1/2 sin3x . e^(2x) - 3/4 . cos3x . e^(2x) - 9/4 "I" + "c"_1`
`"I" + 9/4"I" = 1/2 sin3x . e^(2x) - 3/4 cos3x . e^(2x) + "c"_1`
`13/4 "I" = 1/2 e^(2x) [sin3x - 3/2 cos3x] + "c"_1`
I = `4/13 xx 1/2 e^(2x) [sin3x . 3/2 cos3x] + 4/13 "c"_1 ...[at 4/13 "c"_1 = "c"]`
I = `1/13 e^(2x) [2 sin3x - 3 cos3x] + "c"`
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