Rules of integration

To solve integrals more efficiently, there are several calculation rules similar to those from differential calculus. They make it possible to integrate composite expressions step by step without returning to the definition each time.

 

 

1. Constant factor rule

A constant can always be taken outside the integral sign. If \( \large k \) is a constant and \( \large f(x) \) a function, then:

 

$$ \large \int k \cdot f(x)\,dx \;=\; k \cdot \int f(x)\,dx $$

 

Example:

$$ \large \int 5x^3\,dx \;=\; 5 \cdot \int x^3\,dx \;=\; 5 \cdot \frac{x^4}{4} + C \;=\; \tfrac{5}{4}x^4 + C $$

 

 

2. Sum and difference

Integration distributes over addition and subtraction. This means that each term can be integrated separately:

 

$$ \large \int \big(f(x) \pm g(x)\big)\,dx \;=\; \int f(x)\,dx \;\pm\; \int g(x)\,dx $$

 

Example:

$$ \large \int (3x^2 - 2x + 4)\,dx \;=\; x^3 - x^2 + 4x + C $$

 

 

3. Power rule

The power rule is the most fundamental formula and holds for all \( \large n \neq -1 \):

 

$$ \large \int x^n\,dx \;=\; \frac{x^{n+1}}{n+1} + C $$

 

Example:

$$ \large \int x^4\,dx \;=\; \frac{x^5}{5} + C $$

 

 

4. Logarithmic rule

When the exponent is \( \large -1 \), a special rule applies because the power rule cannot be used. In this case, the natural logarithm appears:

 

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

 

 

5. Exponential functions

The integral of an exponential function with base \( \large e \) is again an exponential function:

 

$$ \large \int e^x\,dx \;=\; e^x + C $$

 

If the exponent is a linear function \( \large ax \), the result is adjusted by \( \large \frac{1}{a} \):

 

$$ \large \int e^{ax}\,dx \;=\; \frac{1}{a}e^{ax} + C $$

 

 

6. Trigonometric functions

The most important integrals of sine and cosine are:

 

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

$$ \large \int \cos x\,dx \;=\; \sin x + C $$

 

 

7. Composite functions (reverse chain rule)

If the function consists of an inner expression whose derivative appears outside, you can integrate “backwards” using substitution (covered later). A simple form is:

 

$$ \large \int f'(x)\,f(x)^n\,dx \;=\; \frac{f(x)^{n+1}}{n+1} + C $$

 

Example:

$$ \large \int 2x(x^2 + 1)^3\,dx \;=\; \frac{(x^2 + 1)^4}{4} + C $$

 

 

Summary

These calculation rules make it possible to find integrals quickly and reliably. The constant factor, sum, and power rules are used in almost every computation. They form the foundation for more advanced methods such as substitution and integration by parts.