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.

Convert the integrand:

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(3) Perform the integral.

Expand integrand and use power rule to obtain:

Insert definition :

This is our final answer.

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 .

Factor out of the integrand:

Swap remaining into type:

Substitute and :

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(3) Compute the integral in and convert back to .

Expand the integrand:

Apply power rule:

Plug back in, :

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) Perform the integral.

Break up the sum:

Observe chain rule backwards:

This is our final answer.