Solve the following integral (showing the steps):

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

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- Oct 24th 2009, 05:10 AMbaseballmanSimple exponential integral
Solve the following integral (showing the steps):

$\displaystyle \int {t{e^t}dt}$ - Oct 24th 2009, 05:28 AMmr fantastic
See - Wolfram|Alpha Click on Show steps.

- Oct 24th 2009, 07:35 AMbaseballman
That's good. But I'm looking for a solution in which integration by parts is NOT used.

- Oct 24th 2009, 11:09 AMKrizalid
- Oct 24th 2009, 01:39 PMmr fantastic
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:

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