Calculus 2 2026-05-19 - Trigonometric Substitution and Integration Techniques

Flashcards covering trigonometric identities, unit circle values, trigonometric substitution rules, and integration strategies for rational functions based on the lecture notes.

Fundamental Pythagorean Identity

$$\sin^2(x) + \cos^2(x) = 1$$

Secondary Pythagorean Identity (Tangent)

$\tan^2(x) + 1 = \sec^2(x)$, obtained by dividing the fundamental identity by $\cos^2(x)$

Reciprocal of Sine

Cosecant ($\csc(x)$), described as sine but flipped

Reciprocal of Cosine

Secant ($\sec(x)$), described as cosine but flipped

Reciprocal of Tangent

Cotangent ($\cot(x)$), described as tangent but flipped

Unit Circle Values at $0$

Cosine is $1$ and sine is $0$

Unit Circle Values at $\pi/6$

Cosine is $\frac{\sqrt{3}}{2}$ and sine is $\frac{1}{2}$

Unit Circle Values at $\pi/4$

Both cosine and sine are $\frac{\sqrt{2}}{2}$

Unit Circle Values at $\pi/3$

Cosine is $\frac{1}{2}$ and sine is $\frac{\sqrt{3}}{2}$

Unit Circle Values at $\pi/2$

Sine is $1$ and cosine is $0$

Trig Substitution for $a^2 + x^2$

Use the substitution $x = a\tan(\theta)$, as anytime there is a plus sign, it involves tangent

Trig Substitution for $a^2 - x^2$

Use the substitution $x = a\sin(\theta)$, used when the constant is positive and the $x^2$ is negative

Trig Substitution for $x^2 - a^2$

Use the substitution $x = a\sec(\theta)$, used when the $x$ is positive and the constant is negative

Power Reduction Formula for Secant

$$\int \sec^n(\theta) \, d\theta = \frac{1}{n-1} \sec^{n-2}(\theta) \tan(\theta) + \frac{n-2}{n-1} \int \sec^{n-2}(\theta) \, d\theta$$

Rational Function Strategy (High Numerator Degree)

When the degree in the numerator is higher than or equal to the degree in the denominator, perform polynomial long division

Partial Fraction Decomposition

A technique to write a complex rational function as a sum of simpler rational functions by factoring the denominator

Irreducible Quadratic Factor

A factor in the denominator like $x^2 + 25$ that has complex solutions and does not factor further into real linear terms, often leading to an arc tangent ($\arctan$) result

Double Angle Formulas (Integration)

Used to integrate $\sin^2(\theta)$ or $\cos^2(\theta)$ by removing the power to make them linear trigonometric functions