Math 175
Notes and Learning Goals
Trigonometric Integration Tricks
Lesson 7-3
Today’s problems explore some trigonometric integration tricks. All of the integrals are standard
examples or things that come up in physical applications. However, the tricks for integrating them
are less commonly used than the “big three” of substitution, integration by parts, and partial
fractions. You should work all of these integrals by hand, without using a calculator or other
machine assistance. The first three types below are covered in Section 7.2 of your textbook. The
last one is in Section 7.3.
1. Substitutions where u = sin x or u = cos x creates a convenient du.
Z
Example:
sin2 x cos x dx. Here u = sin x creates du = cos x dx, which is very convenient.
2. Trig identities to make powers of sine and cosine into nice problems.
Example: Even powers use half angle identities
Z
Z 1 1
2
+ cos(2x) dx
cos x dx =
2 2
PatrickJMT video. (Somewhat messy example.)
Example: Odd powers convert to Type 1.
Z
Z
Z
3
2
cos x dx = cos x cos x dx = (1 − sin2 x) cos x dx
PatrickJMT video.
3. Oscillators of with two different frequencies.1 Again, using a trig idendity.
Example:
Z Z
sin(7x) cos(4x) dx =
1
1
sin(11x) + sin(3x)
2
2
PatrickJMT video.
1
These integrals are a key element of Fourier transforms and Fourier series.
1
dx
4. Trig substitution. This is a lot like u-substitution, but with a sort of backwards twist.
Example:
Z √
1 − x2 dx
Solution: Begin with
x = sin u, then
dx = cos u du, etc....
This is explored in more detail in the exercises.
Khan Academy video.
Here is a full list of the trig identities that could come up in this work.
• Pythagorean
sin2 x + cos2 x = 1
1 + tan2 x = sec2 x
• Half-angle
1 1
− cos(2x)
2 2
1 1
cos2 x = + cos(2x)
2 2
sin2 x =
• Product-to-sum
1
1
cos(a − b) − cos(a + b)
2
2
1
1
cos a cos b = cos(a − b) + cos(a + b)
2
2
1
1
sin a cos b = sin(a + b) + sin(a − b)
2
2
sin a sin b =
2