Solve the following integral (showing the steps):
$\displaystyle \int {t{e^t}dt}$
See - Wolfram|Alpha Click on Show steps.
Consider $\displaystyle y = t \sin t$.
Then $\displaystyle \frac{dy}{dt} = \sin t + t \cos t$.
Now integrate both sides:
$\displaystyle y = \int \sin t + t \cos t \, dt = \int \sin t \, dt + \int t \cos t \, dt $.
Substitute $\displaystyle y = t \sin t$:
$\displaystyle t \sin t = \int \sin t \, dt + \int t \cos t \, dt $.
Re-arrange to make the required integral the subject:
$\displaystyle \int t \cos t \, dt = t \sin t - \int \sin t \, dt = t \sin t + \cos t + C$.