Question

Evaluating the integral

\(\int {t{e^{ - 3t}}} dt\)

Short answer

We should use integration by parts to solve the given integral.

Let, u=t and dv=e^-3tdt

Solve for du and v, and substitute in

\(\int {udv = uv - \int {vdu} } \)

Step-by-step solution

Given Data

Given- I=\(\int {t{e^{ - 3t}}} dt\)

Let, u=t\(\int {dv = } \int {t{e^{ - 3t}}} dt\)

Du=dt \(v = \frac{{ - {e^{ - 3t}}dt}}{3}\)

Integrating the equation

\(\int {t{e^{ - 3t}}} dt\)=\(\frac{{ - t{e^{ - 3t}}}}{3} - \int {\frac{{ - {e^{ - 3t}}}}{3}} dt\)

=\(\frac{{ - t{e^{ - 3t}}}}{3} - \int {(\frac{{ - {e^{ - 3t}}}}{{3( - 3)}})} + c\)

=\(( - t + \frac{1}{3})\frac{{{e^{ - 3t}}}}{3} + c\)

Hence, \(\int {t{e^{ - 3t}}} dt\)= \(( - t + \frac{1}{3})\frac{{{e^{ - 3t}}}}{3} + c\)