Integration

Last updated: December 2024

Disclaimer: These are my personal notes compiled for my own reference and learning. They may contain errors, incomplete information, or personal interpretations. While I strive for accuracy, these notes are not peer-reviewed and should not be considered authoritative sources. Please consult official textbooks, research papers, or other reliable sources for academic or professional purposes.

1. Definition of Integral

The definite integral of $f(x)$ from $a$ to $b$ is:

\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x

where $\Delta x = \frac{b-a}{n}$ and $x_i = a + i\Delta x$.

2. Fundamental Theorem of Calculus

Part 1: If $F(x) = \int_a^x f(t) \, dt$, then $F'(x) = f(x)$.

Part 2: If $F'(x) = f(x)$, then:

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

3. Basic Integration Rules

Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
Constant Rule: $\int c \, dx = cx + C$
Constant Multiple: $\int cf(x) \, dx = c\int f(x) \, dx$
Sum Rule: $\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$

4. Common Integrals

Exponential: $\int e^x \, dx = e^x + C$
Natural Log: $\int \frac{1}{x} \, dx = \ln|x| + C$
Trigonometric:
- $\int \sin(x) \, dx = -\cos(x) + C$
- $\int \cos(x) \, dx = \sin(x) + C$
- $\int \sec^2(x) \, dx = \tan(x) + C$
Inverse Trig:
- $\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C$
- $\int \frac{1}{1+x^2} \, dx = \arctan(x) + C$

5. Integration Techniques

5.1 Substitution

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

\int f(g(x))g'(x) \, dx = \int f(u) \, du

5.2 Integration by Parts

For functions $u(x)$ and $v(x)$:

\int u(x)v'(x) \, dx = u(x)v(x) - \int v(x)u'(x) \, dx

5.3 Partial Fractions

For rational functions, decompose into simpler fractions:

\frac{P(x)}{Q(x)} = \frac{A}{x-a} + \frac{B}{x-b} + \cdots

5.4 Trigonometric Substitution

For integrals involving $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$:

$\sqrt{a^2 - x^2}$: Let $x = a\sin(\theta)$
$\sqrt{a^2 + x^2}$: Let $x = a\tan(\theta)$
$\sqrt{x^2 - a^2}$: Let $x = a\sec(\theta)$

6. Applications

6.1 Area Between Curves

The area between $f(x)$ and $g(x)$ from $a$ to $b$ is:

A = \int_a^b |f(x) - g(x)| \, dx

6.2 Volume of Revolution

For rotation around the $x$-axis:

V = \pi \int_a^b [f(x)]^2 \, dx

6.3 Arc Length

The length of the curve $y = f(x)$ from $a$ to $b$ is:

L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx

7. Improper Integrals

For integrals with infinite limits or unbounded functions:

\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx

The integral converges if the limit exists, diverges otherwise.

8. Numerical Integration

8.1 Trapezoidal Rule

\int_a^b f(x) \, dx \approx \frac{h}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)]

8.2 Simpson's Rule

\int_a^b f(x) \, dx \approx \frac{h}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-1}) + f(x_n)]

9. Code Example

# Python code for integration
import sympy as sp
import numpy as np
from scipy import integrate

# Symbolic integration
x = sp.Symbol('x')
f = x**2 + 2*x + 1
integral = sp.integrate(f, x)
print(f"∫({f}) dx = {integral}")

# Definite integral
definite_integral = sp.integrate(f, (x, 0, 1))
print(f"∫₀¹({f}) dx = {definite_integral}")

# Numerical integration
def func(x):
    return x**2 + 2*x + 1

result, error = integrate.quad(func, 0, 1)
print(f"Numerical result: {result}")

10. References

Stewart, J. (2015). Calculus: Early Transcendentals.
Apostol, T. M. (1967). Calculus, Volume 1.
Spivak, M. (2008). Calculus.