Trigonometric functions, Tangent half-angle substitution, Integration of trigonometric functions, Useful substitutions.
INTEGRATION INVOLVING COMBINATIONS OF TRIGONOMETRIC FUNCTIONS
Integration of trigonometric terms is not easy. We will check out a few simpler tricks and the most important methods. Let's start with one of the most exciting methods.
One of the strangest cases of integration by substitution is when u= tan(x/2).
We use this if sinx and cosx is included in first-degree form in the fraction.
The main idea of the substitution is the following three formulae:
Now we will see some magic tricks.
Those who are not particularly fond of magic, can skip this part.
We start with a smaller trick:
Then, we use a more advanced trick.
At the end comes a finishing trick.
The hocus-pocus ends here, and we got this:
And, one more thing.
For the substitution, we need the derivative of the expressed x, too.
So, we need to express x first.
There is such a thing as:
This way, bye-bye tangent, we found x.
And we found the derivative, too. We know that
The good news is that we had to suffer through this previous ordeal only this one time.
From here, all we have to do is make a note of these, so we could refer to them when needed.
Here comes an exercise.
This method works in more complicated cases, too:
We do some decomposition
and then it's done.
And that's all I have to say about that.