Definition of Integration

The indefinite integral of a function \( f(x) \), denoted as \( \int f(x) \, dx \), is the set of all antiderivatives of \( f(x) \), which means it is the function whose derivative is \( f(x) \). The process of integration is often referred to as "finding the antiderivative." Integration can be thought of as the reverse process of differentiation.

Mathematically, we express this as:

\[ \int f(x)\, dx = F(x) + C \]

Where \( F(x) \) is the antiderivative of \( f(x) \), and \( C \) is the constant of integration, which accounts for the fact that there can be multiple functions that differ by a constant that have the same derivative.

Properties of Indefinite Integrals

1. \[ \int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx \]
2. \[ \int [f(x) - g(x)]\,dx = \int f(x)\,dx - \int g(x)\,dx \]
3. \[ \int k \cdot f(x)\,dx = k \cdot \int f(x)\,dx \quad \text{(where \( k \) is a constant)} \]
4. \[ \int 0\,dx = C \quad \text{(zero function integrates to a constant)} \]
5. \[ \frac{d}{dx} \left( \int f(x)\,dx \right) = f(x) \]
6. \[ \int f(ax + b)\,dx = \frac{1}{a} \int f(u)\,du \quad \text{(where } u = ax + b,\ a \ne 0\text{)} \]

Standard Formulae for Integration

1. \[ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \ne -1) \]
2. \[ \int \frac{1}{x}\,dx = \ln|x| + C \]
3. \[ \int e^x\,dx = e^x + C \]
4. \[ \int a^x\,dx = \frac{a^x}{\ln a} + C \quad (a > 0,\ a \ne 1) \]
5. \[ \int e^{ax + b}\,dx = \frac{1}{a} e^{ax + b} + C \]
6. \[ \int \sin x\,dx = -\cos x + C \]
7. \[ \int \cos x\,dx = \sin x + C \]
8. \[ \int \sec^2 x\,dx = \tan x + C \]
9. \[ \int \csc^2 x\,dx = -\cot x + C \]
10. \[ \int \sec x \tan x\,dx = \sec x + C \]
11. \[ \int \csc x \cot x\,dx = -\csc x + C \]
12. \[ \int \frac{1}{\sqrt{1 - x^2}}\,dx = \sin^{-1} x + C \]
13. \[ \int \frac{1}{1 + x^2}\,dx = \tan^{-1} x + C \]
14. \[ \int \frac{1}{x\sqrt{x^2 - 1}}\,dx = \sec^{-1} |x| + C \]