Integrating: 30 Years Later - No Taylor Book

[denverdoc]

for the life of me I cannot recall how to integrate say

Int(dv/(k-v^2))=dt


Its been 30 years since I took calc, and can't find my Taylor book.

 

[d_leet]

Try the substitution v=k^1/2sin(x)

 

[VietDao29]

for the life of me I cannot recall how to integrate say

Int(dv/(k-v^2))=dt


Its been 30 years since I took calc, and can't find my Taylor book.

Why is there a dt thingy following the integral expression? So, I assume that you want to integrate:
$$\int \frac{dv}{k - v ^ 2}$$
If k > 0, you can do something like this:
$$\int \frac{dv}{k - v ^ 2} = \int \frac{dv}{(\sqrt{k} - v) (\sqrt{k} + v)} = \frac{1}{2 \sqrt{k}} \int \frac{(\sqrt{k} - v) + (\sqrt{k} + v)}{(\sqrt{k} - v) (\sqrt{k} + v)} dv$$
$$= \frac{1}{2 \sqrt{k}} \int \left( \frac{1}{\sqrt{k} + v} + \frac{1}{\sqrt{k} - v} \right) dv = ...$$
Can you go from here? :)
If k < 0 (or in other words, -k > 0), then you can pull out the minus sign, like this:
$$\int \frac{dv}{k - v ^ 2} = - \int \frac{dv}{- k + v ^ 2} = - \int \frac{dv}{(\sqrt{- k}) ^ 2 + v ^ 2} = -\frac{1}{\sqrt{-k}} \arctan \left( \frac{x}{\sqrt{-k}} \right) + C$$, can you get this? :)

 

[denverdoc]

Why is there a dt thingy following the integral expression? So, I assume that you want to integrate:
$$\int \frac{dv}{k - v ^ 2}$$
If k > 0, you can do something like this:
$$\int \frac{dv}{k - v ^ 2} = \int \frac{dv}{(\sqrt{k} - v) (\sqrt{k} + v)} = \frac{1}{2 \sqrt{k}} \int \frac{(\sqrt{k} - v) + (\sqrt{k} + v)}{(\sqrt{k} - v) (\sqrt{k} + v)} dv$$
$$= \frac{1}{2 \sqrt{k}} \int \left( \frac{1}{\sqrt{k} + v} + \frac{1}{\sqrt{k} - v} \right) dv = ...$$
Can you go from here? :)
If k < 0 (or in other words, -k > 0), then you can pull out the minus sign, like this:
$$\int \frac{dv}{k - v ^ 2} = - \int \frac{dv}{- k + v ^ 2} = - \int \frac{dv}{(\sqrt{- k}) ^ 2 + v ^ 2} = -\frac{1}{\sqrt{-k}} \arctan \left( \frac{x}{\sqrt{-k}} \right) + C$$, can you get this? :)

sure--er, I think... in the first case, let u=(sqrt(k)+v)
du=dv then becomes, int(du/u)... =ln(sqrt(k)+v)-ln(sqrt(k)-v)+C=


using deletes hint, v=k^1/2sin:
k^-1/2int(cos(x)/(k(1-sin^2(x)) which can be treated similarly or simply looked up as the integral of secant.

Thanks so much, the memories are coming back.

Can anyone recommend a good calculus text? I still have my differential eqns/engineering math text, just need a good basic text..

 

[VatanparvaR]

Plz, help :

int_{- infinity}^{+ infinity} dk [exp(ikx)]/(k^2+a^2) -?

 

[Gib Z]

Lol mate you can't just find someone else's thread about an integral and put yours in it.

https://www.physicsforums.com/showthread.php?p=1294997#post1294997

Go there, I put your question in the right place.