Review - Module 4: Integration

Flashcards generated from lecture notes on Integration techniques, trigonometric integrals, and integration by parts and partial fractions.

Reverse Power Rule

∫xⁿ dx = (x^(n+1))/(n+1) + C

∫sin(x) dx

-cos(x) + C

∫tan(x) dx

-ln|cos(x)| + C = ln|sec(x)| + C

∫aˣ dx

(a^x)/ln(a) + C

Integrals of the form ∫f(ax+b) dx

If ∫f(x) dx = F(x) + C, then ∫f(ax+b) dx = (1/a)F(ax+b) + C

Integration by Substitution - Idea

Introduce a new variable to convert a hard integral into an easy one

The Substitution Rule

∫g'(x)f(g(x)) dx = ∫f(u) du where u = g(x)

Trigonometric Substitution

Transforming integrals containing √(a² - x²) using the substitution x = a sin θ and the identity 1 - sin² θ = cos² θ.

Type 1 Trigonometric Integrals

∫cosᵐ(x)sinⁿ(x) dx for m, n ∈ Z⁺ ∪ {0}

Type 2 Trigonometric Integrals

∫cosᵐ(x)sinⁿ(x) dx where either m or n (but not both) must be odd

Type 3 Trigonometric Integrals

∫cos(mx)sin(nx) dx, ∫sin(mx)sin(nx) dx, ∫cos(mx)cos(nx) dx for m, n ∈ R

Identity cos²(x) + sin²(x) = 1

sin²(x) = 1 - cos²(x) and cos²(x) = 1 - sin²(x)

sin²(x)

(1 - cos(2x))/2

cos²(x)

(1 + cos(2x))/2

Product Identity for sin(x)cos(y)

sin(x)cos(y) = ½[sin(x+y) + sin(x-y)]

Product Identity for cos(x)cos(y)

cos(x)cos(y) = ½[cos(x+y) + cos(x-y)]

Product Identity for sin(x)sin(y)

sin(x)sin(y) = ½[cos(x-y) - cos(x+y)]

Even Function - cos(x) example

cos(y) = cos(-y)

Odd Function - sin(x) example

-sin(x) = sin(-x)

Integration by Parts Formula

∫udv = uv - ∫vdu

LIATE - Order of preference for choosing 'u' in Integration by Parts

Logs, Inverse Trig, Polynomials, Trig, Exponentials

Double Angle Identity

sin(2θ) = 2sin(θ)cos(θ)

Integration by Partial Fractions

Decompose rational functions into simpler fractions that can be easily integrated.

Proper Rational Function

The degree of the numerator is less than the degree of the denominator.

Improper Rational Function

The degree of the numerator is greater than or equal to the degree of the denominator.

Irreducible Quadratic Factor

A quadratic with no real roots.

Standard Integral

∫ dx / (a² + x²) = (1/a) tan⁻¹ (x/a) + C