How to Simplify the Solution of a Definitive Integral?

[Djokara]

Any ideas how to solve this
$$\int_0^h {\frac{1}{\sqrt{1+(r({\frac{1}{z}}-{\frac{1}{h}}))^2}}}\,dz$$
Don't have an idea from where to begin

 

[SteamKing]

The correct term is 'definite integral'.

Is r a constant?

 

[Djokara]

Yeah, this is problem from ED R is radius and h is height of cone.

 

[LCKurtz]

Any ideas how to solve this
$$\int_0^h {\frac{1}{\sqrt{1+(r({\frac{1}{z}}-{\frac{1}{h}}))^2}}}\,dz$$
Don't have an idea from where to begin

Start by simplifying the integrand.

 

[jackmell]

Any ideas how to solve this
$$\int_0^h {\frac{1}{\sqrt{1+(r({\frac{1}{z}}-{\frac{1}{h}}))^2}}}\,dz$$
Don't have an idea from where to begin

How about this: We have the expression:

$$\frac{1}{\sqrt{1+(r/z-a)^2}}$$

now, can you simplify that and get:

$$\frac{z}{\sqrt{Q(z)}}$$

where $Q(z)$ is a quadratic polynomial in z? Then we'd have:

$$\int \frac{z}{\sqrt{Q(z)}} dz$$

Now I don't know about you, but I'd look in my Calculus textbook about integrands with radicals with quadratic expressions (I did). And what is the first thing done when that happens?

 

[Djokara]

That worked but solution is messy.