Question

Differentiate: 1/(2x-1)^(1/2) dx from x=5 to x=13

Answer 1

Use a u sub, and then F(13)-F(5)

Answer 2

if you can differentiate
\[\int\limits 1/x dx\]
then you will know how to do it.
do you need more help ?

Answer 3

Yes I do, sorry.

Answer 4

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 ?

Answer 5

log x +c?

Answer 6

it's actually lnx + C

Answer 7

That's what I meant, lol

Answer 8

okay just making sure

Answer 9

I just really don't understand definite integrals.

Answer 10

now your goal is to make the integral look like that by letting
u = 2x+1

Answer 11

oops, never mind about lnx, i did not see the ^1/2 part.

so it will be a family of \[\int\limits 1/\sqrt(x)\]

Answer 12

But then what?

Answer 13

after letting u = 2x-1, \[\int\limits 1/\sqrt(u) dx\]

Answer 14

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 ?

Answer 15

No

Answer 16

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 ?

Answer 17

Yes

Answer 18

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

Answer 19

Gotcha

Answer 20

so once you substitute dx with the above equation,

your integral becomes\[1/2\int\limits 1/\sqrt(u) du\]

Answer 21

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.

Answer 22

so the answer is 2?

Answer 23

\[1/2\int\limits 1/\sqrt(u) du = 1/2 *1/2*(u^{1/2})+C\]

Answer 24

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

Answer 25

woops\[1/4 \sqrt(2x-1) +C\]

is what I meant

Answer 26

it seems like the answer will be 3/4

Answer 27

Oh, ok, I see. Thanks!!! Can you help me with a couple word problems too?

Answer 28

I need to go soon, so I can help you with one of them.
ask me the one that you think need most help

Answer 29

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?

Answer 30

okay so what do you not get from this problem ?