What, exactly, are you trying to say? Yes is the general solution to the integration. But what does "for C" mean?
As a result of looking for the analytical solution to the following indefinite integral
I came to the following expression:
for being the integration constant. I am trying now to get the expression of and I am not sure about the following steps:
1) Multiply both terms by -1 (not sure if this is valid with logarithmic expressions):
2) Apply exponentiation to both sides of the previous expression:
Since is a constant ( ), I am not sure how to proceed further. , I would try something like:
But this is not correct. I would appreciate if somebody would give a hint.
Thank you in advance.
Hello, jguzman!
If we have an arbitrary constant, and multiply by -1,
. . we get , another arbitrary constant.
We might as well call it
Then we raise to the power
. . We get: , another arbitrary constant.
We might as well call it
Get the idea?
I came to the following expression: .
Multiply by
Exponentiate: .
We have: .
Therefore: .
Hi everybody!
You guys helped me a lot!. I figured out how to reach to the solution. I'll post it here just in case somebody need it!
From the original expression:
by separating the variables we have:
with substitution of , I get that
To obtain the following expression to be solved by direct integration:
whose solution is:
for being the integration constant. Thanks Prove It for your excellent post about it!
Now it's simple to follow Soroban:
, this was the difficult step for me :S
, now let
Thanks so much !!!
Wov, this now explains several things. I was "having the feeling" that the sign of integration constant, while not being so relevant for the whole arithmetic, may be important for finding the solution, and was wondering how this would affect the resulting expression.
Now, the correct solution would be...
Due to the nature of the model, , then having the negative sing for makes sense in that context. It's so good to have the mathematical rigor!.
Again and again, many thanks!