anonymous
  • anonymous
Show that the function F of x equals the integral from 2 times x to 5 times x of 1 over t dt is constant on the interval (0, +∞).
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
https://gyazo.com/e5425cec35c6e331d016d9664296b104
anonymous
  • anonymous
@terenzreignz @zepdrix I used the fundamental theorem of calc to find derivative which was equal to 0.
anonymous
  • anonymous
That's enough to prove it right?

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terenzreignz
  • terenzreignz
Use the second Fundamental Theorem instead.
anonymous
  • anonymous
Yes That's what was done :p
anonymous
  • anonymous
2nd, 1st it's all the same it's not a race ;)
anonymous
  • anonymous
Slope=0 means no change in function
Owlcoffee
  • Owlcoffee
It is not the same, the second part of the fundamental theorem for integral calculus is known also as Barrows rule, models the following equality: \[\int\limits_{a}^{b}f(x)dx=F(b)-F(a)\]
terenzreignz
  • terenzreignz
I suppose I could start over, like I usually do :/ \[\Large \int\limits_{2x}^{5x}\frac1t dt \]
terenzreignz
  • terenzreignz
And yes, did you use the FTC as Owlcoffee stated it above ^ ?
anonymous
  • anonymous
Yes
anonymous
  • anonymous
I used
terenzreignz
  • terenzreignz
Show me >:)
anonymous
  • anonymous
to be exact
anonymous
  • anonymous
\[F'(x)=u'(x)f(u(x))-v'(x)f(v(x))\]
anonymous
  • anonymous
\[\int\limits_{v(x)}^{u(x)}f(t)dt\]
terenzreignz
  • terenzreignz
It seems I may have to explain the second FTC after all :)
anonymous
  • anonymous
O_o why?
anonymous
  • anonymous
My teacher gave me that
anonymous
  • anonymous
Is it the way I wrote it? Cause I was too lazy to write out the complete thing
anonymous
  • anonymous
It has d/dx in front of the integral since that's what you need to do in this case
terenzreignz
  • terenzreignz
Well, *I* am giving this to you now: Suppose you have a function F(x) such that the derivative of F is f on a given interval. That is: F'(x) = f(x) Then \[\Large \int\limits_a^b f(x) = F(a) - F(b)\] Is the gist of it.
anonymous
  • anonymous
Find the derivative.
anonymous
  • anonymous
Ok
terenzreignz
  • terenzreignz
Things don't seem to be going my way ^^ \[\Large \int\limits_a^b f(x)dx = F(b) - F(a)\]
terenzreignz
  • terenzreignz
There, use that. Now, can you integrate 1/t?
anonymous
  • anonymous
ln(5x)-ln(2x)?
terenzreignz
  • terenzreignz
That's right ^^ Simplify it using the rules concerning logarithms: I'd tell you which rule exactly, but maybe you already know :D
anonymous
  • anonymous
A refresher would be nice
anonymous
  • anonymous
Wait
anonymous
  • anonymous
Just remembered
anonymous
  • anonymous
\[\ln \frac{ 5x }{ 2x }\]?
terenzreignz
  • terenzreignz
Yup. Simplify further?
anonymous
  • anonymous
ln (5/2)
terenzreignz
  • terenzreignz
Which is constant. Et voila ^^
anonymous
  • anonymous
So is what I did considered wrong?
terenzreignz
  • terenzreignz
I'm not exactly sure what you did :> But you do have that tendency to overcomplicate things sometimes :D But that which you used looked nothing like what was needed in this particular question... unless I'm mistaken.
anonymous
  • anonymous
My teacher complicates it not me :p I'm using her formula.
terenzreignz
  • terenzreignz
Well, that's why I'm here :D
anonymous
  • anonymous
https://gyazo.com/cb41c86a344c6fb6dbadfadabd886845
anonymous
  • anonymous
That's what I was given.
anonymous
  • anonymous
I used a variant of the 1st which may seem complicated but it was plug and play as well.
terenzreignz
  • terenzreignz
Part II comes to mind...
anonymous
  • anonymous
It seems simpler but I don't feel like erasing and rewriting
anonymous
  • anonymous
So I think I'm good with what I wrote
anonymous
  • anonymous
:)

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