Find the inverse function

[dandrew7]

Hi, I have a polar curve, r = theta.
One can find the arc length through integration, eventually coming up with:

What I would like to do however, is find the upper bound "b" given lower bound "a" and an arc length.
So for example, if I want an arc length of 5, and my lower bound is pi/2, I'd like to know what the upper bound should be:

But of course, I'd like to make the arc length anything I desire, along with the lower bound.

After rewriting it.. I believe this can be solved by finding the inverse for the following:

"y" is everything that can be punched into a calculator, so its a "known" variable.

What I tried
I tried to get rid of the natural log:
e^y = (sqrt(x^2+1) +x) * e^(xsqrt(x^2+1)) but now I can't get rid of the e^(..) term..
I tried wolfram to solve it for me.. I tried mathway.. I tried symbolab.. none can solve it.

I'm having a little bit of trouble with this though and I'm not the best with logs.. any general pointers?
Thank you!

 

[Steven G]

Usually you can't solve this type of equation for x (where x is both in the base and in the exponent).

 

[Deleted member 4993]

Hi, I have a polar curve, r = theta.
One can find the arc length through integration, eventually coming up with:

What I would like to do however, is find the upper bound "b" given lower bound "a" and an arc length.
So for example, if I want an arc length of 5, and my lower bound is pi/2, I'd like to know what the upper bound should be:

But of course, I'd like to make the arc length anything I desire, along with the lower bound.

After rewriting it.. I believe this can be solved by finding the inverse for the following:

"y" is everything that can be punched into a calculator, so its a "known" variable.

What I tried
I tried to get rid of the natural log:
e^y = (sqrt(x^2+1) +x) * e^(xsqrt(x^2+1)) but now I can't get rid of the e^(..) term..
I tried wolfram to solve it for me.. I tried mathway.. I tried symbolab.. none can solve it.

I'm having a little bit of trouble with this though and I'm not the best with logs.. any general pointers?
Thank you!

As far as I can tell, roots of the equation above can only be estimated through numerical methods (Newton-Raphson, etc.)

 

[dandrew7]

Thank you! I guess I haven't learned those numerical methods in class before, or if I have, I don't remember anyway.
I needed to solve this problem for game development, and the newton raphson method seems to work like a charm.
Only a few iterations are needed to produce a pretty precise answer.