1. Why Integration Rules Are Helpful
Integration rules make it easier to compute antiderivatives without relying on the definition each time. They act like shortcuts that help break down functions into simpler pieces. In personal-notes style: once these rules are known, most integrals turn into quick patterns.
2. Constant Multiple Rule
A constant can be taken outside the integral. This keeps the work simple.
2.1. Formula
\int c \cdot f(x) \, dx = c \int f(x) \, dx
2.2. Example
\int 7 \sin x \, dx = 7(-\cos x) + C
3. Sum and Difference Rule
Integrals can be split across addition and subtraction. This helps handle long expressions term by term.
3.1. Formula
\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx
3.2. Example
\int (x^2 + 5x) \, dx = \frac{x^3}{3} + \frac{5x^2}{2} + C
4. Power Rule for Integrals
The power rule is one of the most used rules. It works for all powers except −1.
4.1. Formula
\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \ne -1)
4.2. Example
\int x^4 dx = \frac{x^5}{5} + C
5. Integral of 1/x
Since the power rule fails for n = −1, this special rule is used.
5.1. Formula
\int \frac{1}{x} dx = \ln|x| + C
5.2. Example
\int \frac{3}{x} dx = 3 \ln|x| + C
6. Integral of Exponential Functions
Exponential functions have clean antiderivatives.
6.1. Formulas
\int e^x dx = e^x + C
\int a^x dx = \frac{a^x}{\ln a} + C
6.2. Example
\int 5^x dx = \frac{5^x}{\ln 5} + C
7. Integral of Basic Trigonometric Functions
These standard trig integrals appear often.
7.1. Formulas
\int \sin x \, dx = -\cos x + C
\int \cos x \, dx = \sin x + C
\int \sec^2 x \, dx = \tan x + C
\int \csc^2 x \, dx = -\cot x + C
\int \sec x \tan x \, dx = \sec x + C
\int \csc x \cot x \, dx = -\csc x + C
8. Combining Rules
Real integrals often mix substitution, algebraic simplification, and basic rules. These basic rules form the toolbox that most integrals rely on.