Calculus II Study Notes: Integration Techniques, Antiderivatives, and Applications

Basic Integration Rules and Antiderivatives

Common Indefinite Integrals

Indefinite integrals, or antiderivatives, are fundamental in calculus for finding functions whose derivatives yield the given integrand. Below are key formulas:

• Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$

• Reciprocal Rule: $\int \frac{dx}{x} = \ln|x| + C$

• Exponential Rule: $\int e^x dx = e^x + C$

• Trigonometric Rules:

  • $\int \cos x dx = \sin x + C$

  • $\int \sin x dx = -\cos x + C$

  • $\int \sec^2 x dx = \tan x + C$

  • $\int \sec x \tan x dx = \sec x + C$

  • $\int \frac{dx}{1+x^2} = \arctan x + C$

  • $\int \frac{dx}{\sqrt{1-x^2}} = \arcsin x + C$

Linearity of Integration: Integration is linear, meaning:

• $\int (af(x) + g(x)) dx = a \int f(x) dx + \int g(x) dx$

Basic Functions and Their Properties

Graphs and Limiting Behavior

Understanding the behavior and graphs of basic functions is essential for calculus. Below are some key functions:

• Cosine Function: $y = \cos x$

  • Period: $2\pi$

  • Range: $[-1, 1]$

  • Key points: $(0,1)$, $(\pi, -1)$, $(2\pi, 1)$

• Sine Function: $y = \sin x$

  • Period: $2\pi$

  • Range: $[-1, 1]$

  • Key points: $(0,0)$, $(\pi/2, 1)$, $(\pi, 0)$, $(3\pi/2, -1)$, $(2\pi, 0)$

• Exponential Growth: $y = e^x$

  • $e^0 = 1$

  • $\lim_{x \to -\infty} e^x = 0$

  • $\lim_{x \to \infty} e^x = \infty$

• Exponential Decay: $y = e^{-x}$

  • $e^0 = 1$

  • $\lim_{x \to \infty} e^{-x} = 0$

• Logarithmic Growth: $y = \ln x$

  • $\ln 1 = 0$

  • $\lim_{x \to 0^+} \ln x = -\infty$

Integration Techniques

Substitution (u-Substitution)

Substitution is used to simplify integrals by changing variables. The method is especially useful when the integrand contains a function and its derivative.

• Let $u = g(x)$, then $du = g'(x) dx$

• Rewrite the integral in terms of $u$ and $du$

• After integrating, substitute back to the original variable

Example: $\int \sin(3x) dx$ Let $u = 3x$, $du = 3dx$, so $dx = du/3$ $\int \sin(3x) dx = \int \sin(u) \frac{du}{3} = \frac{1}{3} \int \sin(u) du = -\frac{1}{3} \cos(u) + C = -\frac{1}{3} \cos(3x) + C$

Integration by Parts

Integration by parts is based on the product rule for differentiation and is used for integrals involving products of functions.

• Formula: $\int u dv = uv - \int v du$

• Choose $u$ and $dv$ such that $du$ and $v$ are easily computable

Example: $\int x e^x dx$ Let $u = x$, $dv = e^x dx$ $du = dx$, $v = e^x$ $\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C$

Partial Fraction Decomposition

Partial fractions are used to break down rational functions into simpler fractions that can be integrated individually.

• If $f(x) = \frac{P(x)}{Q(x)}$ is a proper rational expression, decompose $f(x)$ into a sum of simpler fractions.

• Each factor in $Q(x)$ generates a term in the decomposition.

Type of Factor	Form of Decomposition
Linear $(x-a)$	$\frac{A}{x-a}$
Repeated Linear $(x-a)^n$	$\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}$
Irreducible Quadratic $(x^2 + bx + c)$	$\frac{Bx + C}{x^2 + bx + c}$

Example: $\frac{2x+1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$

Applications of Integration

Definite Integrals and Area

Definite integrals are used to compute the area under a curve between two points, as well as other physical quantities such as total energy consumption or mass.

• $\int_a^b f(x) dx$ gives the net area under $f(x)$ from $x=a$ to $x=b$

• Units of the definite integral are the product of the units of $f(x)$ and $x$

Example: If $r(t)$ is a rate of energy consumption in units of kW, then $\int_0^{24} r(t) dt$ gives total energy consumed in kWh over 24 hours.

Physical Applications: Mass by Integration

Integration can be used to find the mass of a planet with variable density:

• Density function: $\rho(r) = \rho_0 e^{-kr}$

• Mass: $M = \int_0^R 4\pi r^2 \rho(r) dr$

Advanced Integration Techniques

Trigonometric Integrals

Integrals involving trigonometric functions often require identities or substitutions.

• Half-angle identities:

  • $\sin^2 x = \frac{1}{2}(1 - \cos(2x))$

  • $\cos^2 x = \frac{1}{2}(1 + \cos(2x))$

• Substitution for integrals like $\int \sin(ax) dx$ or $\int \cos(ax) dx$

Integration by Substitution and Parts: Strategy Table

Integral	Suggested Method
$\int x^2 \sin(x^2) dx$	Substitution: $u = x^2$
$\int \arctan(x) dx$	Integration by Parts: $u = \arctan(x)$, $dv = dx$
$\int \frac{1}{1+x^2} dx$	Basic Antiderivative: $\arctan(x)$

Summary Table: Common Antiderivatives

Function	Antiderivative
$x^n$	$\frac{x^{n+1}}{n+1} + C$
$\frac{1}{x}$	$\ln|x| + C$
$e^x$	$e^x + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\frac{1}{1+x^2}$	$\arctan x + C$
$\frac{1}{\sqrt{1-x^2}}$	$\arcsin x + C$

Key Concepts and Strategies

• Antiderivatives are unique up to a constant: If $F(x)$ is an antiderivative of $f(x)$, then $F(x) + C$ is also an antiderivative for any constant $C$.

• Choosing the right technique: Use substitution when the integrand contains a function and its derivative; use integration by parts for products of functions; use partial fractions for rational functions.

• Physical interpretation: Definite integrals often represent total quantities, such as area, mass, or energy, depending on the context.

Example Application: To find the total energy consumed over a day, integrate the rate function $r(t)$ over the interval $[0,24]$.

Additional info: These notes summarize key integration techniques, basic function properties, and applications as covered in a standard Calculus II course. For more advanced topics, such as improper integrals or differential equations, consult further chapters.