integral from 0 to 8 of (5xe^-x)(dx)

so far i think

5(e^-x)

so then do i plug in 8

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- May 6th 2010, 04:31 AMcummings15liate rule i think
integral from 0 to 8 of (5xe^-x)(dx)

so far i think

5(e^-x)

so then do i plug in 8 - May 6th 2010, 04:46 AMtom@ballooncalculus
Just in case a picture helps...

Trial and error will quickly establish (and liate confirm) that you need the legs-crossed version of...

http://www.ballooncalculus.org/asy/prod.png

... the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x.

The general drift is...

http://www.ballooncalculus.org/asy/maps/parts.png

In this case...

http://www.ballooncalculus.org/asy/parts/xe.png

And the rest...

__Spoiler__:

_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote! - May 6th 2010, 04:57 AMcummings15
so is it

(5x)(e^-x) + (5) (e^-x) + C - May 6th 2010, 05:02 AMtom@ballooncalculus
The negative of that, as you can maybe see above now that I've fixed the second pic (in the spoiler).

I think you missed that we need the chain rule for e^(-x). I didn't bother zooming in on that, but (gimme a minute)...

http://www.ballooncalculus.org/asy/parts/xe2.png

... where

http://www.ballooncalculus.org/asy/chain.png

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

I uncrossed the legs for clarity - i.e. switched the 5x and e^-x.

_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote! - May 6th 2010, 08:33 AMSoroban
Hello, cummings15!

Quote:

$\displaystyle 5\int^8_0 xe^{-x}\,dx$

. . $\displaystyle \begin{array}{cccccccc}u &=& x && dv &=& e^{-x}\,dx \\

du &=& dx && v &=& -e^{-x} \end{array}$

We have: .$\displaystyle 5\bigg[-xe^{-x} + \int e^{-x}\,dx\bigg] \;=\;5\bigg[-xe^{-x} - e^{-x}\bigg] + C \;=\;-5e^{-x}(x+1) + C$

Evaluate: .$\displaystyle -\frac{5}{e^x}(x+1)\,\bigg]^8_0 \;=\;\bigg[-\frac{5}{e^8}(8+1) \bigg] - \bigg[-\frac{5}{e^0}(0+1)\bigg] \;=\;5 - \frac{45}{e^8} $

- May 6th 2010, 08:55 AMskeeter
tabular integration ...

sign ........... u ........... dv

+............... 5x .......... $\displaystyle e^{-x}$

- ................ 5 .......... $\displaystyle -e^{-x}$

................... 0 .......... $\displaystyle e^{-x}$

$\displaystyle

\left[-5xe^{-x} - 5e^{-x}\right]_0^8

$

$\displaystyle

\left[-5e^{-x}(x+1) \right]_0^8

$

$\displaystyle

5 -\frac{45}{e^8}

$