Theory

Theory 1

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 and thus we can do:

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 and :

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.

Theory 2

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 -substitution using:

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 factor is left behind.

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 or or , among others.

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.