Add Integrals Calculator

What is Integral Calculus?

Integral calculus is one of the two major branches of calculus, along with differential calculus. While differential calculus deals with rates of change and slopes of curves, integral calculus focuses on accumulation of quantities and areas under curves.

The integral of a function represents the area under the curve of that function. This concept has applications in physics, engineering, economics, and many other fields.

Types of Integrals

There are two main types of integrals:

1. Indefinite Integrals: These represent the antiderivative of a function and include a constant of integration (C).
2. Definite Integrals: These calculate the exact area under a curve between two specified limits.

Indefinite integrals are used to find general solutions to differential equations, while definite integrals are used to compute areas, volumes, and other physical quantities.

Basic Integration Rules

Here are some fundamental integration rules that this calculator can compute:

• ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C (for n ≠ -1)
• ∫ eˣ dx = eˣ + C
• ∫ sin(x) dx = -cos(x) + C
• ∫ cos(x) dx = sin(x) + C
• ∫ sec²(x) dx = tan(x) + C

These basic rules form the foundation for more complex integral calculations.

How to Use This Calculator

1. Select whether you want to compute an indefinite or definite integral.
2. Enter the function you want to integrate in the function field.
3. If computing a definite integral, enter the lower and upper limits.
4. Click "Calculate" to see the result.
5. Review the step-by-step solution and formula used.

Example Calculations

Example 1: Indefinite Integral

Compute ∫ 3x² dx

Using the power rule: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C

∫ 3x² dx = 3*(x³/3) + C = x³ + C

Example 2: Definite Integral

Compute ∫[0 to 2] x² dx

First find the antiderivative: ∫ x² dx = (x³)/3 + C

Evaluate at the bounds: [(2³)/3] - [(0³)/3] = (8/3) - 0 = 8/3 ≈ 2.6667