Simple exponential integral

• Oct 24th 2009, 06:10 AM
baseballman
Simple exponential integral
Solve the following integral (showing the steps):

$\int {t{e^t}dt}$
• Oct 24th 2009, 06:28 AM
mr fantastic
Quote:

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

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

See - Wolfram|Alpha Click on Show steps.
• Oct 24th 2009, 08:35 AM
baseballman
That's good. But I'm looking for a solution in which integration by parts is NOT used.
• Oct 24th 2009, 12:09 PM
Krizalid
Quote:

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.
• Oct 24th 2009, 02:39 PM
mr fantastic
Quote:

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

Consider $y = t \sin t$.

Then $\frac{dy}{dt} = \sin t + t \cos t$.

Now integrate both sides:

$y = \int \sin t + t \cos t \, dt = \int \sin t \, dt + \int t \cos t \, dt$.

Substitute $y = t \sin t$:

$t \sin t = \int \sin t \, dt + \int t \cos t \, dt$.

Re-arrange to make the required integral the subject:

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