Indefinite Integrals: Concepts & Properties

Concept of Indefinite Integral

• Integration is the inverse process of derivation, aimed at finding functions $F(x)$ whose derivative results in a given function $f(x)$.
• A primitive of $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$.
• An indefinite integral represents the set of all primitives of a function $f(x)$, denoted as $\int f(x) dx = F(x) + C$.
  • $\int$ Symbolizes integration.
  • $f(x)$ Is the integrand.
  • $dx$ Indicates the variable of integration.
  • $C$ Is the constant of integration.
  • $F(x)$ Is the primitive function.
• Functions with primitives have infinite primitives, differing by a constant: $[F(x) + C]' = F'(x) = f(x)$.
• $\int 5x^4 dx = x^5 + C$ is an example
• $\int cos(x) dx = sin(x) + C$ is an example
• $\int 5 dx = 5x + C$ is an example

Properties of Indefinite Integral

• The integral of a sum is the sum of the integrals: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$.
• The integral of a constant times a function is the constant times the integral: $\int k f(x) dx = k \int f(x) dx$.

Definition of the Natural Logarithm Function

• $ln$ is defined on $]0; +\infty[$ and is the primitive of the function $x \mapsto \frac{1}{x}$ that vanishes at $1$.

Properties of the Natural Logarithm Function

• $ln(1) = 0$
• $ln(e) = 1$
• $ln(ab) = ln(a) + ln(b)$ for $a, b > 0$.
• $ln(a^n) = n \cdot ln(a)$ for $a > 0$ and integer $n$.
• $ln(\frac{a}{b}) = ln(a) - ln(b)$ for $a, b > 0$.
• $ln(\frac{1}{a}) = -ln(a)$ for $a > 0$.
• $e^{ln(x)} = x$ for $x > 0$.
• $ln(e^x) = x$ for all real $x$.

Function study of the Natural Logarithm

• $ln$ function is differentiable on $]0; +\infty[$ and its derivative is $x \mapsto \frac{1}{x}$.
• The natural logarithm is strictly increasing on $]0; +\infty[$.
• $\lim\limits_{x \to 0^+} ln(x) = -\infty$
• $\lim\limits_{x \to +\infty} ln(x) = +\infty$

Curve representation of the Natural Logarithm

• Starts from negative infinity as x approaches 0 from the positive side.
• Crosses the x-axis at x = 1.
• Passes through the point (e, 1) or (2.718, 1).
• Rises at a decreasing rate as x increases.

Derivatives of Natural Logarithm

• If $u$ is a differentiable and strictly positive function on an interval $I$, then $ln(u)$ is differentiable on $I$ and $(ln(u))' = \frac{u'}{u}$.

Limits of Natural Logarithm

• $\lim\limits_{x \to +\infty} \frac{ln(x)}{x} = 0$
• $\lim\limits_{x \to 0} xln(x) = 0$
• $\lim\limits_{x \to 0} \frac{ln(1+x)}{x} = 1$
• $\lim\limits_{x \to +\infty} ln(1 + \frac{1}{x}) = 0$
• $\lim\limits_{h \to 0} \frac{ln(a+h) - ln(a)}{h} = \frac{1}{a}$

Solving equations and inequalities related to logarithms

• $ln(x) = a \Leftrightarrow x = e^a$
• $ln(x) < a \Leftrightarrow 0 < x < e^a$
• $ln(x) > a \Leftrightarrow x > e^a$
• $ln(x) = ln(y) \Leftrightarrow x = y$
• $ln(x) < ln(y) \Leftrightarrow 0 < x < y$
• $ln(x) > ln(y) \Leftrightarrow x > y$

Function study examples

• For $f(x) = ln(x^2 - 1)$, the domain is $x \in ]-\infty; -1[ \cup ]1; +\infty[$.
• The derivative of $f(x) = ln(x^2 - 1)$ is $f'(x) = \frac{2x}{x^2 - 1}$.
• The function $f(x) = ln(x^2 - 1)$ decreases from $-\infty$ to $-1$ and increases from $1$ to $+\infty$.