A Level maths Integration Question

We are currently doing A level practice papers for the Edexcel spec and I came across an 8 marker relating to integration and I have no idea how to approach it. The question goes:
Given that ∫ (top limit lnb. bottom limit LN2) (e^2x/e^2x -1)dx = ln4, find the value of b showing each step in your working.

Any suggestions would be appreciated 😀

Are you familiar with the fact that: $\displaystyle \int \dfrac{f'(x)}{f(x)} .dx = \ln |f(x)| + \mathrm{const.}$ ??

Unfortunately im not familiar with that notation. what does it mean?

I see you're using the TSR app to post comments. If you see code in my last post (and naturally it doesn't make sense) then view this page from a browser instead. The app doesn't translate code at the moment.

Anyway, all it says is that if you are integrating a quotient where the function in the numerator is exact derivative of the function in the denominator, then the integral simply evaluates to log of that denominator function.

Ah, that must be it lol. Okay thank you very much, I'll try and give this a go I've already worked out the integral just need to get b.

My suggestion was a start on doing that, but if you've already done that, what do you get after you work out the integral?

1/2×ln |e^2x -1|

Sounds good, so now just apply the limits.

The upper limit yields: $\dfrac{1}{2}\ln|e^{2\ln b} - 1|$

The lower limit yields: $\dfrac{1}{2} \ln |e^{2\ln 2} - 1|$


These simplify a bit considering that $e^{\ln A} = A$

Okay I've done that I got a decimal value for the Ln2 limit but with regards to the lnb im not sure what to do.

Decimal value...?? No, don't round anything here.

You need to say: $\dfrac{1}{2} \ln |e^{2\ln 2} - 1| = \dfrac{1}{2} \ln 3$

And similarly with the upper limit.

Are you familiar with the fact that: $\displaystyle \int \dfrac{f'(x)}{f(x)} .dx = \ln |f(x)| + \mathrm{const.}$ ??

Maybe you could've been more clear about this 5 years ago. TSR people are notoriously unhelpful. Where and when is this used.