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How to Integrate Tangent/Secant Problems with an Odd, Positive Power of Tangent

Here’s how you integrate a trig integral that contains tangents and secants where the tangent power is odd and positive. You’ll need the tangent-secant version of the Pythagorean identity:

image0.png
  1. Lop off a secant-tangent factor and move it to the right.

    First, rewrite the problem:

    image1.png
  2. Convert the remaining (even) tangents to secants with the tangent-secant version of the Pythagorean identity.

    image2.png

    Now make the switch.

    image3.png
  3. Solve with the substitution method with u = sec(x) and du = sec(x) tan(x)dx.

    image4.png
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