Examples

Power product - odd power

Compute the integral:

Solution

(1) Swap over the even bunch:

Max even bunch leaving power-one is .

Apply to in the integrand:

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(2) Perform -substitution on the power-one integrand:

Set . Hence . Recognize this in the integrand and convert:

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(3) Integrate using power rule:

Insert definition :

Power product - tan and sec

Compute the integral:

Solution

(1) 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.

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(2) Try again: :

Factor out of the integrand:

Swap remaining into type:

Substitute and :

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(3) Integrate in and convert back to :

Trig power product - differing frequencies

Compute the integral:

Solution

(1) Convert product to sum using trig identity:

Use with and :

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(2) Integrate:

Break up the sum:

Observe chain rule backwards: