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Question
Integrate the following with respect to x:
`"e"^(- 4x) sin 2x`
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Solution
`int "e"^("a"x) sin "b"x "d"x = "e"^("a"x)/("a"^2 + "b"^2) ["a" sin "b"x - "b" cos "b"x] + "c"`
Here a = – 4, b = 2
`int "e"^(- 4x) sin 2x "d"x = "e"^(- 4x)/((- 4)^2 + 2^2) [-4 sin 2x - 2 cos 2x] + "c"`
`int "e"^(- 4x) sin 2x "d"x = "e"^(- 4x)/(16 + 4) [- 4 sin 2x - 2 cos 2x] + "c"`
`int "e"^(- 4x) sin 2x "d"x = - "e"^(- 4x)/20 xx 2[2 sin 2x + cos 2x] + "c"`
`int "e"^(- 4x) sin 2x "d"x = - "e"^(- 4x)/10 [2 sin 2x + cos 2x] + "c"`
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