5. Integration: Other Trigonometric Forms
by M. Bourne
We can use the trigonometric identities that we learned earlier to simplify the integration process.
The main identities are shown here for reference:
The process that we use involves using the trigonometric ratios to simplify the expression, or to get the expression into a form that can be integrated.
To integrate a product of powers of sine and cosine, we use
if at least one of the powers is odd.
Example 1: Integrate: 
.
We use
or
if the power of sinx or cosx is even.
Example 2: Integrate: 
.
Example 3: Integrate: 
.
Application - Root Mean Square Value
The root mean square value of the function y with respect to x is given by:
where T is the period of y.
(See Period of Sine and Cosine if you are not sure about this.)
A common use of this concept is effective current. This is the value of the direct current that would produce the same quantity of heat energy in the same time as a certain alternating current. It is used in the design of heaters.
RMS Example 1: Find the root mean square (rms) value of i = 3 + 2 cos t.
Here is the LiveMath solution to this problem.
RMS Example 2: For a current 
show that the root-mean-square of the current for one period
is 
.
Exercises
Integrate each of the given functions:
1. 
2. 
3. 
4. 
5. 
Application - Length of a Curve
The length s of the arc of a curve y = f(x) from x = a to x = b is given by:
Find the length
of the curve y = ln (cos x) from
x = 0 to 
.
Here is the LiveMath solution to this problem.
Find your integral using Mathematica!
Enter multiply using *, square root of x using Sqrt[x] and trigonometry like Sin[x]. See the full list of how to enter math.
Click the blue button to find your integral.
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