Can I apply The Constant Multiple Rule to
and put it outside of the problem, then integrate the remaining parts?
I'm really having difficulty integrating this, everytime I start I just ending up with workings that go on forever, and using Mathcad is just giving me useless answers.
Can anybody please give me some tips, rules, any help at all.
That actually makes a bit more sense to me now to be honest, it's similar to a previous part of the question. I think I might not have to show all the workings, just how to get to a derivative from a previous question, and then say (and since we know from the previous question . . .).
Many thanks Skeeter, I'll continue trying to manipulate things to get there.
One way to integrate involves solving a first-order linear ordinary differential equation. First, note that
That, of course, is just the quotient rule for derivatives. You can integrate it once to obtain
Now, if you could get the integrand to look like the integrand I just mentioned, you'd be done. Let's say you write
All I've done is write the exponential in the denominator, and then multiplied top and bottom by the new denominator, because I want to get a in the denominator. So now I want
This forces my quotient rule to look like this:
Equating the numerator of this RHS with the previous numerator of (1) yields the first-order linear ordinary differential equation
The solution to this DE is
Hence, the integration result is
as WolframAlpha yields.
Or, just pummel (1 - 2 cos x)(3 + 2 cos x) untill you can see the derivative of 1/2 x + cos x.
Get sin squared from cos squared...
Difference of two squares...
There's our derivative...
So we can try and integrate
by parts. We might even see that the other integrand...
... is going to come in handy soon enough. In other words, it's a good example of problems that will yield to integration by parts but dazzle just because they contain both lower forks of the product rule...
... instead of just one. (Key in spoiler...)
Just in case a picture helps with the parts...
.. or else, zooming in on the chain rule...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!