1. [SOLVED] Can someone give me a start integrating this please?

$\displaystyle \int \frac{1}{10 - kv^2} \, dv$

2. Hi Bucephalus

If k is a constant, you can try :

1. Let : $\displaystyle v=\frac{\sqrt{10}}{\sqrt{k}} \sin \theta$

or

2. Partial fraction by first factoring the denominator

Originally Posted by songoku
Hi Bucephalus

If k is a constant, you can try :

1. Let : $\displaystyle v=\frac{\sqrt{10}}{\sqrt{k}} \sin \theta$

or

2. Partial fraction by first factoring the denominator
Hi Songoku

Two things:
1) I want to understand how you got that expression in your first helper.
2) I have done partial fraction decomposition before, but I don't understand what you're saying "factoring the denominator"
$\displaystyle \frac{1}{10} - \frac{1}{(kv)v}$

I'm sorry, I'm really having trouble understanding.
bucephalus

4. Originally Posted by Bucephalus
[snip]
2) I have done partial fraction decomposition before, but I don't understand what you're saying "factoring the denominator"
$\displaystyle \frac{1}{10} - \frac{1}{(kv)v}$

I'm sorry, I'm really having trouble understanding.
bucephalus
Factorise $\displaystyle 10 - kv^2$.

And if you're studying at the level of integral calculus I would sincerely hope that you know that $\displaystyle \frac{1}{10 - kv^2}$ is certainly NOT equal to $\displaystyle \frac{1}{10} - \frac{1}{(kv)v}$ !!!

5. Hi Bucephalus
I want to understand how you got that expression in your first helper
It's because I want to use trigonometry identity : $\displaystyle 1- \sin^2 \theta = \cos^2 \theta$

Using $\displaystyle v=\frac{\sqrt{10}}{\sqrt{k}} \sin \theta$ will change the denominator to :

$\displaystyle 10-kv^2=10-10 \sin^2 \theta=10(1-\sin^2 \theta)=10 \cos^2 \theta$

In fact, you also can use : $\displaystyle \frac{\sqrt{10}}{\sqrt{k}} \cos \theta$ because it will be relevant to identity : $\displaystyle 1- \cos^2 \theta = \sin^2 \theta$

If the question turns out to be $\displaystyle 10+kv^2$ , you can use substitution : $\displaystyle v=\frac{\sqrt{10}}{\sqrt{k}} \tan \theta$ because it is relevant to identity : $\displaystyle 1+\; tan^2 \theta = \sec^2 \theta$

For the second one, follow Mr. F suggestion :
Factorise $\displaystyle 10 - kv^2$

6. Factorising

so factorise this.
$\displaystyle 10 - kv^2$

The only way I can think of to factorise this is:
$\displaystyle 10(1-\frac{k}{10}v^2)$

Is this what you mean?

7. $\displaystyle 10-kv^2 = \sqrt{10}^2-\sqrt{k}^2v^2$

now using the difference of two squares $\displaystyle a^2-b^2 = (a-b)(a+b)$

You get

$\displaystyle 10-kv^2 = \sqrt{10}^2-\sqrt{k}^2v^2 = \sqrt{10}^2-(\sqrt{k}v)^2 = (\sqrt{10}-\sqrt{k}v)(\sqrt{10}+\sqrt{k}v)$

So now $\displaystyle \frac{1}{10-kv^2} = \frac{A}{\sqrt{10}-\sqrt{k}v}+\frac{B}{\sqrt{10}+\sqrt{k}v}$

Find A and B and the integral will be alot easier in this form.

8. Thanks Pickslides

That's an awesome response.