Free Integral Calculator
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Reviewed by Calculator Editorial Team
Integrals are fundamental in calculus for finding areas under curves, volumes, and solving differential equations. This free integral calculator computes both definite and indefinite integrals with step-by-step solutions and visual graphing.
What is an Integral Calculator?
An integral calculator is an online tool that computes integrals of functions. It handles both definite integrals (with limits) and indefinite integrals (antiderivatives). The calculator provides:
- Step-by-step solutions for educational purposes
- Visual graphing of functions and their integrals
- Support for common functions and special cases
- Clear explanation of the results
Key Concepts
Integrals represent accumulation of quantities. Definite integrals calculate the exact area under a curve between two points, while indefinite integrals find the family of functions whose derivative is the original function.
How to Use This Calculator
- Enter the function you want to integrate in the input field
- For definite integrals, specify the lower and upper limits
- Select the variable of integration (usually x)
- Click "Calculate" to compute the integral
- Review the result, solution steps, and graph
Basic Integral Rules
∫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
Formula Used
The calculator uses numerical integration methods for definite integrals and symbolic computation for indefinite integrals. For definite integrals:
Definite Integral Formula
∫[a,b] f(x) dx ≈ Σ f(xᵢ) Δx
Where Δx = (b - a)/n and xᵢ = a + iΔx
For indefinite integrals, the calculator applies integration rules and simplification techniques to find the antiderivative.
Worked Examples
Example 1: Definite Integral
Calculate ∫[0,2] x² dx
- Identify the function: f(x) = x²
- Set limits: a = 0, b = 2
- Apply the formula: ∫[0,2] x² dx = (2³/3) - (0³/3) = 8/3 ≈ 2.6667
Example 2: Indefinite Integral
Find ∫x³ dx
- Apply the power rule: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C
- Substitute n = 3: ∫x³ dx = (x⁴)/4 + C
| Type | Notation | Interpretation | Example |
|---|---|---|---|
| Definite Integral | ∫[a,b] f(x) dx | Area under curve between a and b | ∫[0,π] sin(x) dx = 2 |
| Indefinite Integral | ∫f(x) dx | Family of antiderivatives | ∫x² dx = x³/3 + C |
Frequently Asked Questions
What types of integrals can this calculator solve?
This calculator can solve both definite and indefinite integrals for a wide range of functions, including polynomials, trigonometric functions, exponential functions, and more.
How accurate are the results?
The calculator uses precise numerical methods for definite integrals and exact symbolic computation for indefinite integrals. Results are accurate within the limits of floating-point arithmetic.
Can I use this calculator for homework?
Yes, this calculator is an excellent educational tool. It provides step-by-step solutions that can help you understand the integration process and verify your work.
What if the calculator can't solve my integral?
If the calculator encounters an integral it can't solve, it will display an error message. You may need to simplify the expression or try a different approach.