Integrate with Respect to X Calculator

What is integration?

Integration is a fundamental concept in calculus that represents the accumulation of quantities. It has two main types:

• Definite integral: Calculates the area under a curve between two points.
• Indefinite integral: Finds the antiderivative of a function, representing the family of curves that have the given function as their derivative.

Integration is widely used in physics, engineering, economics, and many other fields to model continuous changes and accumulations.

How to use this calculator

To use the calculator:

1. Enter the function you want to integrate in the "Function" field. Use x as the variable.
2. For definite integrals, enter the lower and upper limits in the "From" and "To" fields.
3. Click "Calculate" to compute the integral.
4. View the result and the visualization of the function and its integral.

Formula used

The calculator uses the trapezoidal rule for numerical integration with 1000 intervals for accuracy.

Worked examples

Example 1: Definite Integral

Find the area under the curve of f(x) = x² from x = 0 to x = 2.

Using the calculator:

• Function: x^2
• From: 0
• To: 2

The result is approximately 2.6667, which matches the exact value of (2³)/3 - (0³)/3 = 8/3 ≈ 2.6667.

Example 2: Indefinite Integral

Find the antiderivative of f(x) = 3x² + 2x + 1.

Using the calculator:

• Function: 3x^2 + 2x + 1
• Leave "From" and "To" empty for indefinite integral

The result is x³ + x² + x + C, which is the correct antiderivative.