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W02 - Examples

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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.

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.

07 - Trig power product - differing frequencies

Compute the integral:

Solution

Convert product to sum using trig identity.

Use with and :

Perform the integral.

Break up the sum:

Observe chain rule backwards:

This is our final answer.

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: