Calculus 2 2026-05-18 - Trigonometric Substitution Lecture

A set of vocabulary flashcards covering the concepts, substitutions, and identities used in the Trigonometric Substitution method for calculus integration.

Pythagorean Theorem

A mathematical principle used in calculus to relate the sides of a right triangle, identifying that if one side is $x$ and the hypotenuse is $9$, the remaining side is $\text{sqrt}(9 - x^2)$.

Domain of $x$ for $\text{sqrt}(a^2 - x^2)$

The set of values where a substitution like $x = 3\text{sin}(\theta)$ is valid; for $\text{sqrt}(9 - x^2)$, $x$ must be between $-3$ and $3$ to avoid complex numbers.

Intermediate Value Theorem

A theorem mentioned as a hidden component in the substitution process because the range for the function $\text{sin}(\theta)$ remains between $-1$ and $1$.

Sine Substitution (Case 1)

A method where $x = a\text{sin}(\theta)$ and $dx = a\text{cos}(\theta)d\theta$ are used to simplify integrals involving the radical expression $\text{sqrt}(a^2 - x^2)$, utilizing the identity $1 - \text{sin}^2(\theta) = \text{cos}^2(\theta)$.

Power Reduction Formula

A trigonometric identity used to integrate squared trig functions; for example, $\text{cos}^2(\theta) = \frac{1}{2}(1 + \text{cos}(2\theta))$ and for sine, it involves a minus sign: $\text{sin}^2(\theta) = \frac{1}{2}(1 - \text{cos}(2\theta))$.

Reference Triangle

A geometric tool built to convert trigonometric expressions (like $\text{cos}(\theta)$) back into the original variable $x$ after the integration is complete.

Cosecant Anti-derivative

An elementary anti-derivative that involves the natural log, often written in a form similar to the anti-derivative of secant: $\text{ln}|\text{csc}(\theta) - \text{cot}(\theta)|$.

Tangent Substitution (Case 2)

A method used for integrals involving $\text{sqrt}(x^2 + a^2)$ where $x = a\text{tan}(\theta)$ and $dx = a\text{sec}^2(\theta)d\theta$, converting the expression into a form of $\text{sec}(\theta)$, since $1 + \text{tan}^2(\theta) = \text{sec}^2(\theta)$.

Secant Substitution (Case 3)

A substitution used when the integral contains $\text{sqrt}(x^2 - a^2)$, setting $x = a\text{sec}(\theta)$ and $dx = a\text{sec}(\theta)\text{tan}(\theta)d\theta$.

Rationalizing the Denominator

A step in the simplification process where square roots are removed from the denominator by multiplying the numerator and denominator by that root.

Definite Integral Bounds Conversion

The process of changing the original limits of integration (in terms of $x$) to new limits (in terms of $\theta$) using the substitution formula before evaluating the integral.