Examples

Trig sub in quadratic - completing the square

Compute the integral:

Solution

(1) Complete the square to obtain Pythagorean form:

Add and subtract to get desired constant term:

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(2) Perform shift substitution:

Set as inside the square:

Infer . Plug into integrand:

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(3) Trig sub with :

Use substitution . (From triangle or memorized tip.)

Infer . Plug in data:

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(4) Integrate using ad hoc memorized formula:

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(5) Convert trig back to :

First in terms of , referring to the triangle:

Then in terms of using . Plug everything in:

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(6) Simplify using log rules:

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: