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: .

Answer

We use

if the power of sinx or cosx is even.

Example 2: Integrate: .

Answer

Example 3: Integrate: math expression .

Answer

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.

LIVEMath

Answer

RMS Example 2: For a current show that the root-mean-square of the current for one period is .