Trig power products
Videos, Math Dr. Bob:
Videos, Organic Chemistry Tutor:
01 Theory
Review: trig identities
Trig power product:
A
power product has this form: for some integers
and (even negative!). To compute these integrals, use a sequence of these techniques:
- Swap an even bunch.
-sub for power-one. - Power-to-frequency conversion.
Memorize these three techniques!
Examples of trig power products:
Swap an even bunch
If either
or is an odd power, use (maybe repeatedly) to convert an even bunch to the opposite trig type.
An even bunch is all but one from the odd power.
For example:
-sub for power-one If
or , perform -substitution to do the integral. The other trig power becomes a
power; the power-one becomes .
For example, using
By combining these tricks you can do any power product with at least one odd power! Make sure to leave a power-one from the odd power when swapping an even bunch.
Notice:
, even powers. So the method works for and similar.
Power-to-frequency conversion
Using these ‘power-to-frequency’ identities (maybe repeatedly):
change an even power (either type) into an odd power of cosine.
For example, consider the power product:
You can substitute appropriate powers of
By doing some annoying algebra, this expression can be expanded as a sum of smaller powers of
Each of these terms can be integrated by repeating the same techniques.
02 Illustration
Example - Trig power product with an odd power
05 - Power product - odd power
Compute the integral:
Solution
Swap over the even bunch.
Max even bunch leaving power-one is
: Apply to
in the integrand:
Perform
-substitution on the power-one integrand. Set
. Hence
. Recognize this in the integrand. Convert the integrand:
Perform the integral.
Expand integrand and use power rule to obtain:
Insert definition
: This is our final answer.
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03 Theory
Trig power product:
or A
power product has this form: A
power product has this form:
To integrate these, swap an even bunch using:
OR:
Or do
OR:
Note: There is no simple “power-to-frequency conversion” for tan / sec !
We can modify the power-one technique to solve some of these. We need to swap over an even bunch from the odd power so that exactly the
Considering all the possibilities, one sees that this method works when:
is an odd power (with some secants present!) is an even power
Quite a few cases escape this method:
- Any
with no power of - Any
for even and odd
These tricks don’t work for
Special integrals: tan and sec
We have:
These integrals should be memorized individually.
Extra - Deriving special integrals - tan and sec
The first formula can be found by
-substitution, considering that . The second formula can be derived by multiplying
by a special “ ”, computing instead by expanding the numerator and doing -sub on the denominator.
04 Illustration
Example - Trig power product with tan and sec
06 - Power product - tan and sec
Compute the integral:
Solution
Try
. Factor
out of the integrand: We then must swap over remaining
into the type. Cannot do this because
has odd power. Need even to swap.
Try
. Factor
out of the integrand: Swap remaining
into type: Substitute
and :
Compute the integral in
and convert back to . Expand the integrand:
Apply power rule:
Plug back in,
: This is our final answer.
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Trig substitution
Videos, Math Dr. Bob:
- Trig sub 1: Basics and
and and - Trig sub 2:
- Trig sub 3:
- Trig sub 4:
- Trig sub 5:
05 Theory
Certain algebraic expressions have a secret meaning that comes from the Pythagorean Theorem. This meaning has a very simple expression in terms of trig functions of a certain angle.
For example, consider the integral:
Now consider this triangle:
The triangle determines the relation
Now plug these into the integrand above:
Considering that
We must rewrite this in terms of
Here is the moral of the story:
Re-express the Pythagorean expression using a triangle and a trig substitution.
In this way, square roots of quadratic polynomials can be eliminated.
There are always three steps for these trig sub problems:
- (1) Identify the trig sub: find the sides of a triangle and relevant angle
. - (2) Solve a trig integral (often a power product).
- (3) Refer back to the triangle to convert the answer back to
.
To speed up your solution process for these problems, memorize these three transformations:
(1)
(2)
(3)
For a more complex quadratic with linear and constant terms, you will need to first complete the square for the quadratic and then do the trig substitution.
06 Illustration
Example - Trig sub in quadratic: completing the square
08 - Trig sub in quadratic - completing the square
Compute the integral:
Solution Notice square root of a quadratic.
Complete the square to obtain Pythagorean form.
Find constant term for a complete square:
Add and subtract desired constant term:
Simplify:
Perform shift substitution.
Set
as inside the square: Infer
. Plug into integrand:
Trig sub with
. Identify triangle:
Use substitution
. (From triangle or memorized tip.) Infer
. Plug in data:
Compute trig integral.
Use ad hoc formula:
Convert trig back to
. First in terms of
, referring to the triangle: Then in terms of
using . Plug everything in:
Simplify using log rules.
Log rule for division gives us:
The common denominator
can be pulled outside as . The new term
can be “absorbed into the constant” (redefine ). So we write our final answer thus:
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