How to Integrate Compositions of Functions
How to Integrate Rational Expressions Using the Sum, Constant Multiple, and Power Rules
Find the Integral of Nested Functions

How to Integrate Even Powers of Secants with Tangents

You can integrate even powers of secants with tangents. If you wanted to integrate tanm x secn x when n is even — for example, tan8 x sec6 x — you would follow these steps:

  1. Peel off a sec2 x and place it next to the dx:

    image0.png
  2. Use the trig identity 1 + tan2 x = sec2 x to express the remaining secant factors in terms of tangents:

    image1.png
  3. Use the variable substitution u = tan x and du = sec2x dx:

    image2.png

At this point, the integral is a polynomial, and you can evaluate it.

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