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Question
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Solution
Let I = `int e^-x cos 2x.dx`
∫ uv dx = u ∫ v dx – ∫ (u' ∫ v dx) dx.
I = `cos 2x int e^-x dx – int int e^(-x). d/dx cos 2x. dx`
I = `cos 2x. (e^-x)/(d/dx (-x)) – int(e^-x)/(d/dx (- x)). (- sin 2x. d/dx 2x) dx`
I = `- cos 2x. e^-x – int (- e^(-x)) . (- 2sin 2x) dx`
I = `- cos 2x. e^-x – 2 int e^(-x). sin 2x dx`
I = `- cos 2x. e^-x - 2 [sin 2x. int e^-x dx - int int e^(-x) dx. d/dx sin 2x. dx]`
I = `- cos 2x. e^-x - 2 sin 2x. (e^-x)/(- 1) + 2 int (e^-x)/(- 1). cos 2x. 2. dx`
I = `- cos 2x. e^-x + 2 sin 2x. (e^-x) - 2 int 2. e^(-x). cos 2x.dx`
I = `- cos 2x. e^-x + 2 sin 2x. (e^-x) - 4 int e^(-x). cos 2x.dx`
I = `- cos 2x. e^-x + 2 sin 2x. (e^-x) - 4I`
I + 4I = `- cos 2x. e^-x + 2 sin 2x. (e^-x)`
5I = `e^-x (2. sin 2x - cos 2x)`
I = `e^-x/5 (2. sin 2x - cos 2x) + C`
Notes
Let I = `int e^-x cos 2x.dx`
∫ uv dx = u ∫ v dx – ∫ (u' ∫ v dx) dx.
I = `cos 2x int e^-x dx – int int e^(-x). d/dx cos 2x. dx`
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