# Differential calculus

• Dec 19th 2007, 12:14 PM
Rydbirk
Differential calculus
How do I go from

$\displaystyle \int \frac{dv}{cv^2 - g} = \int~dt$

to

$\displaystyle \frac{1}{2\sqrt{cg}}\ln\left(\frac{\sqrt g + v\sqrt c}{\sqrt g - v\sqrt c}\right)=a-t$

It's not just urgent - it's VERY urgent! :/ Big assignment..
• Dec 19th 2007, 12:23 PM
colby2152
Shouldn't it be a + t?
• Dec 19th 2007, 12:25 PM
Rydbirk
Hmm..
My calculator gives me a-t..
• Dec 19th 2007, 01:19 PM
Soroban
Hello, Rydbirk!

Quote:

How do I go from: .$\displaystyle \int \frac{dv}{cv^2 - g} \:= \:\int \,dt$

. . . to: .$\displaystyle \frac{1}{2\sqrt{cg}}\ln\left|\frac{\sqrt g + v\sqrt c}{\sqrt g - v\sqrt c}\right|\;=\;C-t$

The right side is: .$\displaystyle \int dt \;=\;t + C$

Let: $\displaystyle cv^2 \:=\:u^2\quad\Rightarrow\quad v \:=\:\frac{1}{\sqrt{c}}u\quad\Rightarrow\quad dv \:=\:\frac{du}{\sqrt{c}}$
Also let: $\displaystyle g \:=\:a^2$

The left side becomes: .$\displaystyle \int\frac{1}{u^2-a^2}\left(\frac{du}{\sqrt{c}} \right) \;=\;\frac{1}{2a\sqrt{c}}\ln\left|\frac{u-a}{u+a}\right| + C$

Back-substitute: .$\displaystyle u \:=\:v\sqrt{c}\;\text{ and }\; a = \sqrt{g}$

. . $\displaystyle \frac{1}{2\sqrt{g}\sqrt{c}} \ln\left|\frac{v\sqrt{c} - \sqrt{g}}{v\sqrt{c} + \sqrt{g}}\right|$

And we have: .$\displaystyle \frac{1}{2\sqrt{cg}}\ln\left|\frac{v\sqrt{c} - \sqrt{g}}{v\sqrt{c} + \sqrt{g}}\right| \;=\;t + C$