Another math problem. integration

The question goes like this: y = f(x) is a decreasing function in the first quadrant, that passes through (0, 1).For any point P on f(x), the distance between P and the x-intercept of the tangent at P, is 1. Evaluate integration of f(x)dx from 0 to infinity.
I have worked out the relationship between y and x by integration according to the info that the length is 1. But it looks rather complicated (i do not know how to type it out but it is ln(a function of y)+square root of another function of y equals to x) which I have no idea how to further integrate it from 0 to infinity.

Here is a picture of what i have calculated and my problem is how to integrate it further.

Where is the question from (and the other one)?

Sorry for replying late. I do not know. I have after school lessons on math and that teacher gave it to me.

Wbf, it almost sounds like some sort of competition type question.
But you could flip the integration problem / variables here so integrate wrt y and its not too bad so a part circle and some arctrig-type terms.

Wbf, it almost sounds like some sort of competition type question.
But you could flip the integration problem / variables here so integrate wrt y and its not too bad so a part circle and some arctrig-type terms.

Ohno! I met some problems. I am not sure about the upper limit. The lower limit is given in the question that when x=0, y=1. But for the upper limit, i think that the derivative of y wrt x is a function of y, which is also a function of x. Since there is a negative sign in the derivative, as x increases, y will decreases. So as x goes to infinity, y goes to 0. Is that right? And in the question it mentions that in the first quadrant the function is decreasing. So here when i integrate, do i still consider the function is in the first quadrant?
Here os a picture of what i have worked out now.

Its kind of hard to just see some of your working, but the suggestion about flipping the integration variables was based on

So it doesnt matter whenther you consider the usual vertical riemann strips and integrate y(x) wrt x from x = 0 to inf, or consider horizontal reimann strips and integrate x(y) wrt y from y = 0 to 1. The integral isnt that bad and the ans is fairly simple.
But all this really is beyond a level and youre only posting snippets and saying its given by a teacher outside school sounds strange.

Thanks. This question is really hard for me. My teacher may not expect me to work out the right answer. He just wants to give me some challenges and see what my idea is. And let me discuss it with other students. Actually there are a few weird questions he gave to us and i’m not sure.. just to let us think deeper i guess.

Id check with your teacher about asking for hints from others.