Understanding Integrals: Basics, Rules, and Examples

Integration is the process of determining a function from its derivative, representing accumulation or area under a curve. It forms one of the foundational operations in calculus, alongside differentiation.

Mathematical Definition of the Integral Using Riemann Sums

Let $f(x)$ be a real-valued function defined on a closed interval $[a, b]$. Partition $[a, b]$ into $n$ subintervals of width $\Delta x_i = x_i - x_{i-1}$, where $a = x_0 < x_1 < ... < x_n = b$. Select points $\xi_i \in [x_{i-1}, x_i]$ for each subinterval.

The Riemann sum for $f(x)$ over $[a, b]$ is given by

$\displaystyle S = \sum_{i=1}^n f(\xi_i) \Delta x_i$

The definite integral of $f(x)$ over $[a, b]$ is defined as the limit of the Riemann sum as the partition is refined such that the width of the largest subinterval, $\max \Delta x_i$, approaches zero:

$\displaystyle \int_a^b f(x)\,dx = \lim_{\max \Delta x_i \to 0} \sum_{i=1}^n f(\xi_i) \Delta x_i$

Formal Meaning of Indefinite Integral and Antiderivative

Given a function $f(x)$, a function $F(x)$ is called an antiderivative if $\dfrac{d}{dx} F(x) = f(x)$. The indefinite integral of $f(x)$ is denoted by

$\displaystyle \int f(x)\,dx = F(x) + C$

where $C$ is an arbitrary constant of integration. The process of finding such a function $F(x)$ is termed integration.

Fundamental Theorem of Calculus: Complete Statement

The Fundamental Theorem of Calculus connects differentiation and integration in two precise statements:

Part 1: If $f$ is integrable on $[a, b]$ and $F(x) = \int_a^x f(t)\,dt$, then $F'(x) = f(x)$ wherever $f$ is continuous at $x$.

Part 2: If $F$ is any antiderivative of $f$ on $[a, b]$, then

$\displaystyle \int_a^b f(x)\,dx = F(b) - F(a)$

Integration of Standard Functions: Explicit Results

Standard integrals frequently appear in calculus problems. Their evaluation utilises knowledge of derivatives and algebraic manipulation. Below, explicit computations for several canonical forms are provided.

Integral of Power Function: For $n \neq -1$,

$\displaystyle \int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$

Integral of $\frac{1}{x}$:

$\displaystyle \int \frac{1}{x}\,dx = \ln|x| + C$

Integral of Exponential Function:

$\displaystyle \int e^x\,dx = e^x + C$

Integral of Sine and Cosine:

$\displaystyle \int \sin x\,dx = -\cos x + C$

$\displaystyle \int \cos x\,dx = \sin x + C$

Derivation of the Integral of $1/x$

To determine $\int \dfrac{1}{x}dx$, begin with the derivative of the natural logarithm.

$\dfrac{d}{dx} \ln |x| = \dfrac{1}{x}$

Let $F(x) = \ln|x|$, so $F'(x) = \dfrac{1}{x}$

By the definition of antiderivative,

$\int \dfrac{1}{x}dx = \ln|x| + C$

Explicit Calculation: Integral of $\tan x$

Let the objective be to find the indefinite integral of the tangent function, $\int \tan x\, dx$.

Recall that $\tan x = \dfrac{\sin x}{\cos x}$.

Express the integral:

$\int \tan x\,dx = \int \frac{\sin x}{\cos x}\,dx$

Let $u = \cos x$; then $\frac{du}{dx} = -\sin x$ which gives $du = -\sin x\,dx$.

Therefore, $\sin x\,dx = -du$

Substitute into the integral:

$\int \frac{\sin x}{\cos x}\,dx = \int \frac{-du}{u} = -\int \frac{du}{u}$

$\int \frac{du}{u} = \ln|u| + C$

$\int \tan x\,dx = -\ln|u| + C = -\ln|\cos x| + C$

Alternatively, as $\ln \left| \frac{1}{\cos x} \right| = -\ln|\cos x|$, it is sometimes written as

$\int \tan x\,dx = \ln|\sec x| + C$

Explicit Computation: Integral of $\ln x$

Consider the problem $\int \ln x\,dx$ with $x > 0$. Apply integration by parts with $u = \ln x$, $dv = dx$, $du = \dfrac{1}{x}dx$, $v = x$.

By the integration by parts formula,

$\int u\,dv = uv - \int v\,du$

Substituting,

$u = \ln x$, $v = x$, $du = \dfrac{1}{x}dx$

$\int \ln x\,dx = x \ln x - \int x \cdot \frac{1}{x} dx$

$x \cdot \dfrac{1}{x} = 1$

So, $\int x \cdot \frac{1}{x} dx = \int 1\,dx = x$

Hence, $\int \ln x\,dx = x \ln x - x + C$

For advanced integration techniques such as integration by parts, refer to the Integration By Parts resource.

Explicit Computation: Integral of $\sec x$

To compute $\int \sec x\,dx$, multiply numerator and denominator by $\sec x + \tan x$:

$\int \sec x\,dx = \int \frac{\sec x (\sec x + \tan x)}{\sec x + \tan x} dx$

$\sec x (\sec x + \tan x) = \sec^2 x + \sec x \tan x$

So the integral becomes:

$\int \frac{\sec^2 x + \sec x \tan x}{\sec x + \tan x}\,dx$

Let $u = \sec x + \tan x$. Then, $\frac{du}{dx} = \sec x \tan x + \sec^2 x$.

Therefore, $du = (\sec x \tan x + \sec^2 x) dx$

Hence, the numerator equals $du$, and

$\int \frac{du}{u} = \ln|u| + C = \ln|\sec x + \tan x| + C$

Result: $\int \sec x\,dx = \ln|\sec x + \tan x| + C$

For a comprehensive outline of all integration techniques and formulas, consult the Integral Calculus Revision Notes.

Worked Examples: Integration of Standard Functions

Example 1: Evaluate $\int x^2\,dx$.

Substitution: Apply the power rule: $n = 2$

$\int x^2\,dx = \dfrac{x^{2+1}}{2+1} + C = \dfrac{x^3}{3} + C$

Example 2: Compute $\int_0^\pi \sin x\,dx$.

Antiderivative: $\int \sin x\,dx = -\cos x + C$

Definite Integral: $[-\cos x]_{x=0}^{x=\pi} = -\cos \pi - (-\cos 0)$

$\cos \pi = -1$, $\cos 0 = 1$

$-(-1) - (-1) = 1 - (-1) = 2$

Result: $\int_0^\pi \sin x\,dx = 2$

Conceptual Meaning and Applications of the Integral

The definite integral $\displaystyle \int_a^b f(x)\,dx$ quantifies the net area under the curve $y = f(x)$ from $x = a$ to $x = b$. If $f(x)$ represents a rate of change, then the definite integral measures the total accumulation over $[a, b]$. Indefinite integrals provide families of antiderivatives, essential for solving differential equations and expressing general solutions.

For an overview of integral calculus as a whole, refer to the Integral Calculus Overview.