Math question about integrals

[AgaBoogaBoo]

dv=e^(r/2)
v=2e^(r/2)

Going from v to dv easy, it makes sense, but what about the other way around? I understand where e^(r/2) came from, but not where the 2 comes from.

Could someone explain this or tell me what I need to search for? The first example in the wikipedia entry covers just this, but my question is why? Is it a rule you just have to remember? What is happening causing it to be 1/c instead of just c?

 

[Ricemarine]

When you take the anti derivative, you have to divde e^(r/2) by one half, thus, since it's U / (1 / 2), its an improper fraction, and so you must multiply by two hence the antiderivative being v = 2e^(r/2)

r/2 = r * 1/2. since e^r = e^r, you just worry about the 1/2.

 

[AgaBoogaBoo]

Originally posted by: Ricemarine
When you take the anti derivative, you have to divde e^(r/2) by one half, thus, since its U / (1 / 2), its an improper fraction, and so you must multiply by two hence the antiderivative being v = 2e^(r/2)

r/2 = r * 1/2. since e^r = e^r, you just worry about the 1/2.

Why do we divide by one half though? Or is it just whatever e is raised to, is divided by. Example: e^(r/99) would be divided by 1/99.

 

[stinkynathan]

chain rule

 

[Ricemarine]

Originally posted by: stinkynathan
chain rule

I forgot the word for that :laugh:. Basically he's right.

 

[compnovice]

n/m

 

[AgaBoogaBoo]

Originally posted by: Ricemarine

Originally posted by: stinkynathan
chain rule

I forgot the word for that :laugh:.

So then are we finding the derivative of r/2?

 

[Ricemarine]

Originally posted by: AgaBoogaBoo

Originally posted by: Ricemarine

Originally posted by: stinkynathan
chain rule

I forgot the word for that :laugh:.

So then are we finding the derivative of r/2? If so, then this all makes sense.

Believe so...
Basically to prove this with U substitution...

int[e^(r/2)], with U = r/2, du = 1/2dx --> dx = 2du

therefore

int[e^(r/2)] --> int[e^u 2du] --> 2 * int[e^u du] --> [2e^u] -> [2e^(r/2)]
*Since 2 is a constant, we can put it outside the integral.

 

[AgaBoogaBoo]

Originally posted by: Ricemarine

Originally posted by: AgaBoogaBoo

Originally posted by: Ricemarine

Originally posted by: stinkynathan
chain rule

I forgot the word for that :laugh:.

So then are we finding the derivative of r/2? If so, then this all makes sense.

Believe so...
Basically to prove this...

int[e^(r/2)], with U = r/2, du = 1/2dx --> dx = 2du

therefore

int[e^(r/2)] --> int[e^u 2du] --> 2 * int[e^u du] --> [2e^u] -> [2e^(r/2)]
*Since 2 is a constant, we can put it outside the integral.

Wow, don't know why, but I had been skipping that part of it for a very long time and it always bugged me. Thank you very much for writing it out, it's just what I needed. If I can ever help you in some way, drop me a PM and let me know.

 

[Ricemarine]

Originally posted by: AgaBoogaBoo
Wow, don't know why, but I had been skipping that part of it for a very long time and it always bugged me. Thank you very much for writing it out, it's just what I needed. If I can ever help you in some way, drop me a PM and let me know.

It's all good 🙂, ya gave me good info last time :laugh:..

Although I did realized dx was supposed to be dr
Hope it doesn't confuse you 😀

 

[AgaBoogaBoo]

Originally posted by: Ricemarine

Originally posted by: AgaBoogaBoo
Wow, don't know why, but I had been skipping that part of it for a very long time and it always bugged me. Thank you very much for writing it out, it's just what I needed. If I can ever help you in some way, drop me a PM and let me know.

It's all good 🙂, ya gave me good info last time :laugh:..

Although I did realized dx was supposed to be dr
Hope it doesn't confuse you 😀

Nope, got the main point across, which is what I needed 🙂