Integrals and Life
by EpicSkills32, May 10, 2014, 11:08 PM
So here's a nice problem I'm surprised I got on the first try.
![\[ \int_{0}^{49} \, e^{-2\sqrt{x}} \, dx \]](//latex.artofproblemsolving.com/7/a/a/7aa2ca0fa5030a7dd011564ad5cb472e326ed6f0.png)
btw listen to while looking over this post and/or my blog. (I was listening to these and a some other music on air1's new music online channel, and whenever a cool song came on I looked it up and shared it here.) I really encourage you to listen to those songs anytime, especially the first and last.
![\[ \int_{0}^{49} \, e^{-2\sqrt{x}} \, dx \]](http://latex.artofproblemsolving.com/7/a/a/7aa2ca0fa5030a7dd011564ad5cb472e326ed6f0.png)
.
.
.
in and stick a 
at the end, we're gonna have some weird terms to cancel. Let's use 
instead of 
![\[ \int e^{(u)} \cdot (-\sqrt{x}du) \]](http://latex.artofproblemsolving.com/a/6/d/a6d5986dd405ba89480cdffd0676f4719a285c15.png)
(Put parentheses so it's clear what we're substituting; also notice how we'll forget the endpoints and just take it as an indefinite integral)
term and thought I did something wrong. After substituting, any 
.
So... 
.![\[ \int e^{(u)} \cdot (-\sqrt{x}du)=\int e^{u} \cdot \dfrac{1}{2} u \, du \]](http://latex.artofproblemsolving.com/5/6/9/569d9ca7120d109a2bda683dc0ed441ce6cb4dc4.png)
We can pull the 
out of the entire integral:![\[ = \dfrac{1}{2} \, \int e^{u} \, u \, du \]](http://latex.artofproblemsolving.com/8/b/6/8b6b3a23e5ae1a40a02af79916f2fee2a81476a4.png)
and 
.
and 
and 
.
.
is just itself. (The derivative of 
is ![\[ =\dfrac{1}{2}\left(ue^u-e^u\right) \]](http://latex.artofproblemsolving.com/5/f/4/5f489a1c48639db7aa6eb6abd1b6df781bd4430e.png)
Now we can substitute back in for ![\[ = \dfrac{1}{2}\left((-2\sqrt{x})e^{(-2\sqrt{x})}-e^{(-2\sqrt{x})}\right) \]](http://latex.artofproblemsolving.com/e/6/0/e6043493e1c57b03affc7a2e216478b99c0e8b63.png)
![\[ \int e^{-2\sqrt{x}} \, dx = \dfrac{1}{2}\left(-2\sqrt{x}e^{-2\sqrt{x}}-e^{-2\sqrt{x}}\, (+C) \, \right) \]](http://latex.artofproblemsolving.com/4/f/f/4ff23308854a8a9a220a1e5554b712734799ca3a.png)
![\[ =\dfrac{1}{2}\left(-2\left(\sqrt{49}\right)e^{-2\left(\sqrt{49}\right)}-e^{-2\left(\sqrt{49}\right)}\right) - \dfrac{1}{2}\left(-2(0)e^{-2(0)}-e^{-2(0)}\right) \]](http://latex.artofproblemsolving.com/4/6/d/46dbca1834fd6d0aa3dd58878639e29ba1ff07eb.png)
![\[ = \dfrac{1}{2}\left(-15e^{-14}\right)-\dfrac{1}{2}(-1) \]](http://latex.artofproblemsolving.com/c/5/0/c50f1776b95af4cb8b77ba2fa0df1e9d0eb61ada.png)
![\[ = -\dfrac{15}{2e^{14}}-\left(-\dfrac{1}{2}\right) \]](http://latex.artofproblemsolving.com/2/5/8/2583f5aebe8168fb0d1a455361cc5a952e1dff89.png)
![\[ \int_{0}^{49} \, e^{-2\sqrt{x}} \, dx = -\dfrac{15}{2e^{14}}+\dfrac{1}{2} \]](http://latex.artofproblemsolving.com/2/9/4/294b036b4d7d1652802756ba0c953c25e9f97c17.png)
btw listen to while looking over this post and/or my blog. (I was listening to these and a some other music on air1's new music online channel, and whenever a cool song came on I looked it up and shared it here.) I really encourage you to listen to those songs anytime, especially the first and last.
This post has been edited 1 time. Last edited by EpicSkills32, May 10, 2014, 11:13 PM
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