Simple exponential integral

**baseballman** · October 24th 2009, 05:10 AM

Solve the following integral (showing the steps):

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

**mr fantastic** · October 24th 2009, 05:28 AM

Originally Posted by baseballman

Solve the following integral (showing the steps):

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

See - Wolfram|Alpha Click on Show steps.

**baseballman** · October 24th 2009, 07:35 AM

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

**Krizalid** · October 24th 2009, 11:09 AM

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.

**mr fantastic** · October 24th 2009, 01:39 PM

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:

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

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