integral from 0 to 8 of (5xe^-x)(dx)
so far i think
5(e^-x)
so then do i plug in 8
Just in case a picture helps...
Trial and error will quickly establish (and liate confirm) that you need the legs-crossed version of...
... the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x.
The general drift is...
In this case...
And the rest...
Spoiler:
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
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)...
... where
... 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!
Hello, cummings15!
Integrate by parts . . .$\displaystyle 5\int^8_0 xe^{-x}\,dx$
. . $\displaystyle \begin{array}{cccccccc}u &=& x && dv &=& e^{-x}\,dx \\
du &=& dx && v &=& -e^{-x} \end{array}$
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} $
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}
$