🎓 Mathematics • Calculus
100% Client-Side Privacy
Integral Calculator
Computes general antiderivatives F(x) and evaluates definite integrals over integration limits [a, b].
Integrate: ∫ (axⁿ + bx + c) dx
Computes the antiderivative using the reverse power rule and exact definite integral bounds.
Presets:
Limits of Integration:
Definite Integral ∫[0 to 3]
60
Antiderivative F(x)1x^3 + 2x² + 5x + C
Fundamental Theorem F(b) − F(a)60 − (0) = 60
Step-by-Step Integration Derivation
1. Reverse Power Rule: ∫ xⁿ dx = (xⁿ⁺¹) / (n + 1)
2. Term 1: ∫ (3x^2) dx = 3/(2+1) x^3 = 1x^3
3. Term 2: ∫ (4x) dx = 4/2 x² = 2x²
4. Constant term: ∫ (5) dx = 5x
5. Antiderivative: F(x) = 1x^3 + 2x² + 5x + C
6. F(3) = 60
7. F(0) = 0
8. Definite Integral = F(3) − F(0) = 60
What is an Integral?
Integration is the reverse process of differentiation. The definite integral calculates the exact accumulated area under a curve.
The Fundamental Theorem of Calculus states that ∫[a to b] f(x)dx = F(b) - F(a).
Formula & Step-by-Step Calculation
∫ xⁿ dx = (xⁿ⁺¹) / (n + 1) + C, ∫[a to b] f(x)dx = F(b) - F(a)
Reverse power rule for integration.
Worked Step-by-Step Examples
Example 1
∫ (3x² + 4x) dx from 0 to 2
Solution: 16
• F(x) = x³ + 2x². F(2) = 8 + 8 = 16. F(0) = 0. 16 - 0 = 16.
Common Real-World & Academic Use Cases
- ✓ Physics displacement from velocity (x = ∫v dt)
- ✓ Structural center of mass
- ✓ Probability density distribution
How to Use the Integral Calculator
1
Select Definite/Indefinite
Choose with or without integration bounds.
2
Input Coefficients
Enter polynomial powers.
3
Read Integral
Inspect antiderivative F(x) and definite area.
Frequently Asked Questions
Q: What does + C represent?
The arbitrary constant of integration in indefinite integrals.