anonymous
  • anonymous
Differentiate: 1/(2x-1)^(1/2) dx from x=5 to x=13
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
Use a u sub, and then F(13)-F(5)
yuki
  • yuki
if you can differentiate \[\int\limits 1/x dx\] then you will know how to do it. do you need more help ?
anonymous
  • anonymous
Yes I do, sorry.

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yuki
  • yuki
your integral looks awfully like 1/x, it's just shifted a little. you you can guess that it is the family of 1/x. can you integrate 1/x ?
anonymous
  • anonymous
log x +c?
yuki
  • yuki
it's actually lnx + C
anonymous
  • anonymous
That's what I meant, lol
yuki
  • yuki
okay just making sure
anonymous
  • anonymous
I just really don't understand definite integrals.
yuki
  • yuki
now your goal is to make the integral look like that by letting u = 2x+1
yuki
  • yuki
oops, never mind about lnx, i did not see the ^1/2 part. so it will be a family of \[\int\limits 1/\sqrt(x)\]
anonymous
  • anonymous
But then what?
yuki
  • yuki
after letting u = 2x-1, \[\int\limits 1/\sqrt(u) dx\]
yuki
  • yuki
is what you are going to get, but the problem is that the integrand is now a function of u, which cannot be integrated over x. so you have to over come that by figuring out what "dx" is. that technique is the so-called u-substitution. do you know how to find dx ?
anonymous
  • anonymous
No
yuki
  • yuki
okay, so u = 2x-1 right ? if you differentiate both sides with respect to x, d/dx(u) = d/dx(2x-1) d/dx(u) = 2 do you get this so far ?
anonymous
  • anonymous
Yes
yuki
  • yuki
so if you multiply dx on both sides and divide 2 on both sides, you will get 1/2 du = dx this is what is happening, but it is traditional and easier to say u=2x-1 du = 2 dx by differentiating both sides
anonymous
  • anonymous
Gotcha
yuki
  • yuki
so once you substitute dx with the above equation, your integral becomes\[1/2\int\limits 1/\sqrt(u) du\]
yuki
  • yuki
now it is important to know that the limits of the integration will change as well, unless you want to know how to do that I won't go into the detail. to avoid having to change the limits, you will find out the indefinite integral first, then substitute 2x-1 back into u. that way you can plug in the limits again.
anonymous
  • anonymous
so the answer is 2?
yuki
  • yuki
\[1/2\int\limits 1/\sqrt(u) du = 1/2 *1/2*(u^{1/2})+C\]
yuki
  • yuki
1/4(2x-1) + C =F(x) so F(13) -F(5) is your answer. just as a reminder, C will not matter so you can ignore it
yuki
  • yuki
woops\[1/4 \sqrt(2x-1) +C\] is what I meant
yuki
  • yuki
it seems like the answer will be 3/4
anonymous
  • anonymous
Oh, ok, I see. Thanks!!! Can you help me with a couple word problems too?
yuki
  • yuki
I need to go soon, so I can help you with one of them. ask me the one that you think need most help
anonymous
  • anonymous
A manufacturer has been selling flashlights at $6 a piece, and at this price consumers have been buying 3000 flashlights per month. The manufacturer wishes to raise the price and estimates that for each $1 increase in price, 1000 fewer flashlights will be sold each month. The manufacturer can produce the flashlight a cost of $4 per flashlight. At what price should the manufacturer sell the flashlights to generate the greatest profit?
yuki
  • yuki
okay so what do you not get from this problem ?

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