You need to use the substitution -3t = u , such that:
-3t= u => -3dt = du => dt = -(du)/3
Replacing the variable, yields:
int t*e^(-3t) dt = (1/9)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
(1/9)int u*e^u du = (1/9)(u*e^u - int e^u du)
(1/9)int u*e^u du = (1/9)(u*e^u - e^u) + c
Replacing back the variable, yields:
int t*e^(-3t) dt = (1/9)((-3t)*e^(-3t) - e^(-3t)) + c
Hence, evaluating the integral, using substitution, then integration by parts, yields int t*e^(-3t) dt = ((e^(-3t))/9)(-3t - 1) + c
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