1. Simple exponential integral

Solve the following integral (showing the steps):

$\displaystyle \int {t{e^t}dt}$

2. Originally Posted by baseballman
Solve the following integral (showing the steps):

$\displaystyle \int {t{e^t}dt}$
See - Wolfram|Alpha Click on Show steps.

3. That's good. But I'm looking for a solution in which integration by parts is NOT used.

4. Originally Posted by baseballman
That's good. But I'm looking for a solution in which integration by parts is NOT used.
there's no scape dear.

5. Originally Posted by baseballman
That's good. But I'm looking for a solution in which integration by parts is NOT used.
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:

$\displaystyle \int t \cos t \, dt = t \sin t - \int \sin t \, dt = t \sin t + \cos t + C$.