Happy birthday fz!!!
by EpicSkills32, Jan 19, 2015, 1:58 AM
![$\ [\text{Blog Post 145}] $](http://latex.artofproblemsolving.com/3/1/d/31d53e217f4a056b61779d782dc8f8549af1141f.png)
[Somebody halp me!]
Compute:
![$\dfrac{d}{dx} \left[ \int_1^x \dfrac{6}{t^5+3} dt \right] $](http://latex.artofproblemsolving.com/c/b/0/cb0e782c2b8b5dcf5bd5cdbe74fbcd22382623db.png)
So this definitely isn't a normal-looking calculus problem. However, if you understand the Fundamental Theorem of Calculus this should be a piece of cake. . . ok maybe you need to understand derivative and integral function properties sorta too. . .
I was actually paying attention in class, and the teacher did a few example problems similar to this, but for some reason I still don't understand little bits of the steps, hence I cannot complete this problem successfully. I could work it out the way I think it should be and check my answer by approximating with Wolfram but . . . eh.
So here's what I do know: The whole integral inside the derivative brackets is considered a function in this problem. The integral is evaluating the signed area under a curve with parameter t (a defined function of t), and doing so over an interval of 1 to an arbitrary x. Thus this is a function of x. The derivative operator outside is simply taking the derivative of this function.
The fact that we have an integral being evaluated at/to "x" for a differential "t" is kind of confusing. But here's how we think about it:
The whole term
is a function we'll call 
. What's 
? To differentiate with respect to x we need some x but there's only an x in the domain of integration. . .but remember what we figured out above -The integral is telling us signed area for whatever x we put in; there's our function in x.
Because our <uh darn forgot what it's called (index maybe?)> is just to x, the term inside A is just x. In some cases (like the example problem we did in class), it could be something else, even another function (like
) In that case you would use chain rule and whatnot.So what to do now? Well we now have something of the form
![$\dfrac{d}{dx} \left[ A(x)\right] $](http://latex.artofproblemsolving.com/3/a/1/3a19fc7821e374d86d07c0dbe5047e0458891f84.png)
where is a function defined for our purposes.So do I just take the derivative?
I'm confused. . .
![\[ \dfrac{d}{dw}\left[\int_0 ^{\sqrt{w}} \left( \sin^2k+\cos k \right) dk \right] \]](http://latex.artofproblemsolving.com/7/2/c/72c1601c37fd77b75562d8d614253819075c093d.png)
On a completely unrelated note, I'm back in EDM Qluster on plug.dj, the room I started out in when I first joined plug. It's nice to visit old rooms while no one from my regular room is on. XP
This post has been edited 6 times. Last edited by EpicSkills32, Jan 29, 2015, 6:55 AM
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