This part of the course describes how to integrate trigonometric functions, and how to use trigonometric functions to calculate otherwise intractable integrals.
» Session 68: Integral of sinⁿ(x) cosᵐ(x), Odd Exponents
» Session 69: Integral of sinⁿ(x) cosᵐ(x), Even Exponents
» Session 70: Preview of Trig Substitution and Polar Coordinates
» Session 71: Integrals Involving secant, cosecant and cotangent
» Session 72: Trig Substitution
» Session 73: Completing the Square
» Problem Set 9
In this session you will:
Use Integration Techniques (PDF) to do the problems below.
| Section | Topic | Exercises |
|---|---|---|
| 5B | Integration by direct substitution | 9, 11, 13, 16 |
| 5C | Trigonometric integrals | 5, 7, 9, 11 |
| 5D | Integration by inverse substitution | 1, 2, 7, 10 |
Solutions to Integration Techniques problems (PDF)
This problem set is from exercises and solutions written by David Jerison and Arthur Mattuck.
Finding antiderivatives of trigonometric functions can be tricky. In this session, guest lecturer Haynes Miller reviews some trigonometric identities and uses them to get antiderivatives for some products of sine and cosine.
Clip 1: Review of Trigonometric Identities
Clip 2: Integral of sinn(x) cos(x)
Clip 3: Integral of sin3(x) cos2(x)
Integral of cos3(2x)
In this session we complete the search for antiderivatives of products of sine and cosine.
Clip 2: Integral of sin2(x) cos2(x)
Integral of sin4(x) cos2(x)
For some integrals it’s advantageous to use polar coordinates, replacing x by r times the cosine of theta and y by r sine theta.
Clip 1: Area of Part of a Circle
An Alternate Solution
We conclude the discussion of integrating trig functions by reviewing the derivatives of tangent and secant, then finding antiderivatives for these functions and for cosecant.
Clip 1: Review of Trigonometric Identities
Clip 4: Summary of Trig Integration
Integral of tan3(x)
This technique is useful for integrating square roots of sums of squares. This session also covers the trigonometry needed to convert your answer to a more useful form.
Clip 1: Example of Trig Substitution
Clip 2: Undoing Trig Substitution
Clip 3: Summary of Trig Substitution
Substitution Practice
For integrals involving the square root of some more general quadratic function, complete the square and then use trig substitution.
Complete the Square
