Secant and Tangent Integrals

Calculus 2 · Axiom Academy

LESSON Secant and Tangent Integrals One strategy for every : save the right factor for du , then trade secants and tangents with a Pythagorean identity. 1. The Two Derivatives That Power Everything Every technique here rests on two facts. Differentiating hands you a ; differentiating hands you a . Watch each function tip over into its derivative — the piece it produces is exactly the factor you will later hide inside du . 2. The Two Base Integrals (and Secant's Famous Trick) Reversing the derivatives gives and the headline result . The secant one looks like it appears from nowhere — until you watch the trick: multiply by and the numerator becomes exactly the derivative of the denominator. 3. Powers of Tangent: Peel Off For , split one off the front and rewrite it as . That single swap turns the integral into a piece you can u -substitute (with ) plus a lower power of tangent you already know how to handle. Each pass drops the tangent power by 2. Repeat until you reach (odd start) or (even start). 4. Even Powers of Secant: Save One for du When the secant power is even, peel off a single to be your du . Convert every remaining secant to tangent with , substitute , and the integral becomes a polynomial in u — pure power rule. 5. Mixed Powers : Read the Parity Now both appear together. One look at the exponents tells you which factor to save for du and which identity does the converting. The animation walks the single decision that drives every mixed problem.

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