integrate the following
What do you mean "integrate ..."? Are you trying to solve the differential equation? Let . Then you have:
(Note: I factored a -1 from the denominator)
Now, you can integrate both sides. On the left, use partial fractions: .
Once you solve for , plug that in for . (Hint: to solve for v, you will need to use the quadratic formula).
It is a common substitution to try. I am not sure if there is a specific "way to tell". I think practice is your best bet. The more you practice, the more likely you will notice a pattern among problems where that specific substitution works. Perhaps someone else on the forum has a better suggestion.
So, you have:
For the second integral, let . Then you have:
Plugging back in for and using properties of log, you get:
Exponentiating both sides, you get:
Multiplying out, you get:
So, by the quadratic formula:
(If you are having trouble with coefficients to plug into the quadratic formula, , you have )
Plugging that in, you get: which is defined over the reals for all .
You may want to check it out to make sure that it works:
Taking the derivative, you get:
Solving for y', you get , which is not equal to what you started with, so it seems likely that I made a mistake in my calculations somewhere.
Let's try that again:
Let's simplify the constant: let . Then
(That simplifies things greatly).
Let's see what we get from :
So, it actually works out after all (just doesn't look like it will).