(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex]I_{n} = \int^{2}_{0} x^{n}e^{x} dx[/tex] where n is a positive integer. Use integration by parts to show

that

[tex]

2^{n}e^{2} - nI_{n-1}

[/tex]

By first finding

[tex]I_{1} = \int ^{2}_{0} xe^{x} dx [/tex]

find I2 and I3.

2. Relevant equations

I'm sure your all aware of the formula for Int by parts. We'll take the [tex]e^{x}[/tex] function as the one to integrate and the [tex]x^{n}[/tex] as the one to differentiate.

3. The attempt at a solution

So use integration by parts to find In:

[tex]x^{n}e^{x} - n\int^{2}_{0}x^{n-1}[/tex]

Well that's

[tex]x^{n}e^{x} - nI_{n-1}[/tex]

is it not?

So now put the limits in

[tex][x^{n}e^{x} - nI_{n-1}]^{2}_{0}[/tex]

[tex][2^{n}e^{2} - nI_{n-1}] - [ - nI_{n-1}][/tex]

but that's isn't right because I'm getting no [tex] nI_{n-1}[/tex] because they cancel!

Where have I gone wrong?

Thanks

Thomas

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# Homework Help: Reduction formula question (int by parts)

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