Indefinite Integral ; Find the following integral e^(2x) sin(x) dx

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aruv | High School Teacher | (Level 2) Valedictorian

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`I=intsin(x)e^(2x)dx`

`=(1/2)sin(x)e^(2x)-int(1/2)cos(x)e^(2x)dx`

`=(1/2)sin(x)e^(2x)-(1/2){(1/2)cos(x)e^(2x)+int(1/2)sin(x)e^(2x)dx}`

`=(1/4)e^(2x)(2sin(x)-cos(x))-(1/4)intsin(x)e^(2x)dx`

`=(e^(2x)/4)(2sin(x)-cos(x))-(1/4)I`

`I+(1/4)I=(e^(2x)/4)(2sin(x)-cos(x))`

`I((4+1)/4)=(5I)/4=(e^(2x)/4)(2sin(x)-cos(x))`

`Thus`

`I=(e^(2x)/5)(2sin(x)-cos(x))+c`

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