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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