Chapter Contents

• Methods of Integration
• 1. Integration: The General Power Formula
• 2. Integration: The Basic Logarithmic Form
• 3. Integration: The Exponential Form
• 4. Integration: The Basic Trigonometric Forms
• 5. Integration: Other Trigonometric Forms
• 6. Integration: Inverse Trigonometric Forms
• 7. Integration by Parts
• 8. Integration by Trigonometric Substitution
• 9. Integration by Use of Tables
• Table of Common Integrals
• 10. Integration by Reduction Formulae
• 11. Integration by Partial Fractions
• Methods of Integration Problem Solver

• Comments, Questions?

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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

or

if the power of sin x or cos x is even.

Example 2:

Integrate: .

Answer

Example 3:

Integrate: .

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.)

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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.

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RMS Example 1: Find the root mean square (rms) value of i = 3 + 2 cos t.

Answer

RMS Example 2: For a current i given by i = i₀ sin ωt, show that the root-mean-square of the current for one period is .

Answer

Exercises

Integrate each of the given functions:

1.

Answer

2.

Answer

3.

Answer

4.

Answer

5.

Answer

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 .

Answer