Indefinite Integrals

Yaser Rahmati | یاسر رحمتی

Definition

Given a function, $f(x)$, an anti-derivative of $f(x)$ is any function $F(x)$ such that:

$$F'(x)=f(x)$$

If $F(x)$ is any anti-derivative of $f(x)$ then the most general anti-derivative of $f(x)$ is called an indefinite integral and denoted,

$$\int f(x) dx = F(x) + c$$

where $c$ is an arbitrary constant.

In this definition the ∫ is called the integral symbol, $f(x)$ is called the integrand, $x$ is called the integration variable and the “$c$” is called the constant of integration.

Example

What function did we differentiate to get the following function?

$$f(x)=x^2 + 2x + 1$$

In fact, any function of the form,

$$F(x)=\int (x^2 + 2x + 1) dx =\frac{1}{3}x^3+x^2+x+c$$

will give $f(x)$ upon differentiating.

Python

You can use the SymPy library in Python to calculate the integral of a function symbolically. Here’s an example of how you can do this:

import sympy as sp

# Define the symbolic variable
x = sp.symbols('x')

# Define the function symbolically
f = x**2 + 2*x + 1

# Calculate the integral of the function
F = sp.integrate(f, x)

# Print the result
print('The integral of the function f(x) = x^2 + 2x + 1 is:', F)

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