Question

integral from 1 to 2 of 1/(2x -1) dx

Answer 1

Basically is x^2 - x and just substitute 1 and 2 into the function and add them.

Answer 2

my answer was ln(2x-1)/2 then i substituted 1 and 3, dont you have to change the bounds, because you are using u-substitution?

Answer 3

Yes, when you apply a \(u\)-substitution, you will change the bounds according to the choice of \(u\).

Answer 4

Acess denied is that what you get ln(2x-1)/2 for your solution?

Answer 5

Well, when you change the bounds of the integral, you don't need to back-substitute. The bounds are changed to accommodate the variable change to u. If you back-substitute, you'd just use the original bounds for x.
Your antiderivative of 1/(2x - 1) being ln(2x - 1)/2 is correct, though.

Answer 6

thank you.

Answer 7

so...
integral from 1 to 2 of 1/(2x - 1) dx
u = 2x - 1
du = 2 dx --> 1/2 du = dx
= integral from u(1) to u(2) of 1/u * 1/2 du
= 1/2 integral from 1 to 3 of 1/u du
= 1/2 * ln(u) from 1 to 3 ### here we use the bounds 1 and 3 for u.
= 1/2 * ln(3) - 1/2* ln(1)
= 1/2 * ln(3)

Answer 8

why dont you sub u=2x-1 back in?

Answer 9

Well, if we substitute it back in, we'd basically be changing the variable again. We need to change the bounds around to do that.
Notice, however, that our function x of u:
u = 2x - 1
u + 1 = 2x
x = (u + 1)/2
Is the inverse of the u-function.
We change these bounds:
x(1) = (1 + 1)/2
= 1
x(3) = (3 + 1)/2
= 2

You would just get your original boundaries back. There would be no point in that boundary change if we just back-substitute...

Answer 10

I see. So when doing u-substitution, and we change the bounds, DO NOT CHANGE BACK! Leave the bounds the same and sub in for u. Yes?

Answer 11

Yes, you only need to do the u-substitution and change the bounds once. Then the definite integral proceeds as normal

Answer 12

thanks.

Answer 13

You're welcome! :)