`int y/(e^(2y)) dy` Evaluate the integral

Expert Answers

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You need to use the substitution `-2y = u` , such that:

`-2y=  u => -2dy = du => dy= -(du)/(2)`

Replacing the variable, yields:

`int y*e^(-2y) dy = (1/4)int u*e^u du`

You need to use the integration by parts such that:

`int fdg = fg - int gdf`

`f = u => df = du`

`dg = e^u=> g = e^u`

`(1/4)int u*e^u du =(1/4)(u*e^u - int e^u du)`

`(1/4)int u*e^u du = (1/4)u*e^u - (1/4)e^u + c`

Replacing back the variable, yields:

`int y*e^(-2y) dy = (1/4)((-2y)*e^(-2y) - e^(-2y)) + c`

Hence, evaluating the integral, using substitution, then integration by parts, yields `int y*e^(-2y) dy = ((e^(-2y))/4)(-2y - 1) + c`

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