A lecture on trigonometric integrals and trigonometric substitution
1 Trigonometric Integrals
First, we must recall a few trigonometric identities:
and are immediate integrals.
For we use formulas (3) and (4) respectively, e.g.
For integrals of the form or we use substitution with or respectively, e.g.
In the following examples, we use equations (1) in the forms or to transform the integral into one of the type described in Example 1.3.
Similarly one can solve .
In order to solve we express it first as and then proceed as in the previous example.
One can use similar tricks to solve integrals which involve products of powers of and , by using Equation (2). Also, recall that the derivative of is while the derivative of is .
2 Trigonometric Substitutions
One can easily deduce that has value . Why? Simply because the graph of the function is half a circumference of radius (because if you square both sides of you obtain which is the equation of a circle or radius ). Therefore, the area under the graph is a quarter of the area of a circle.
How does one compute without using the geometry of the problem? This is the prototype of integral where a trigonometric substitution will work very nicely. Notice that neither substitution nor integration by parts will work appropriately.
Suppose we want to solve with analytic methods. We will use a substitution (so will be our new variable of integration), because, as we know from Equation (1), , thus getting rid of the pesky square root. Notice that . We also need to find the new limits of integration with respect to the new variable of integration, namely . When we must have . Similarly, when one has . We are now ready to integrate:
Notice that we made use of Equation (4) in the second line.
Similarly, one can solve by using a substitution . Indeed, and . The limits of integration with respect to are again to (check this!). Thus:
Thus, we have proved that a quarter of a circle of radius has area which implies that the area of such a circle is , as usual.
The trigonometric substitutions usually work when
expressions like , ,
appear in the integral at hand, for some real
number . Here is a table of the suggested change of variables
in each particular case:
|If you see…||try this…||because…|
The above are “suggested” substitutions, they may not be the most ideal choice! For example, for the integral , the change will work much better than .
We would like to find the value of
Since neither a -substitution nor integration by parts seem appropriate, we try , . When one has while implies . Hence:
and the last integral is easy to compute using Equation (4).
|Title||A lecture on trigonometric integrals and trigonometric substitution|
|Date of creation||2013-03-22 15:38:39|
|Last modified on||2013-03-22 15:38:39|
|Last modified by||alozano (2414)|