Integral Calculator
Enter Your Integral
Integral Solution
—
Integrals are the inverse of derivatives. The indefinite integral gives a family of antiderivatives. The definite integral computes the area under a curve.
Complete Integral Calculus Guide
Integration is one of the two fundamental operations of calculus, along with differentiation. It's used to find areas, volumes, central points, and many other important quantities.
What is an Integral?
An integral is the inverse of a derivative. If the derivative of F(x) is f(x), then the integral of f(x) is F(x) + C (where C is the constant of integration).
Types of Integrals
| Type | Notation | Description | Result |
|---|---|---|---|
| Indefinite | ∫f(x)dx | No limits, antiderivative | F(x) + C |
| Definite | ∫[a,b]f(x)dx | With limits, area under curve | Number |
| Improper | ∫[a,∞]f(x)dx | Infinite limits | Convergent/Divergent |
| Multiple | ∬f(x,y)dxdy | Multi-variable integration | 3D volume |
Common Integration Rules
| Rule | Formula | Example |
|---|---|---|
| Power Rule | ∫x^n dx = x^(n+1)/(n+1) + C | ∫x² dx = x³/3 + C |
| Sum Rule | ∫[f(x)+g(x)]dx = ∫f(x)dx + ∫g(x)dx | ∫(x+2)dx = x²/2 + 2x + C |
| Constant Multiple | ∫cf(x)dx = c∫f(x)dx | ∫5x dx = 5x²/2 + C |
| Exponential | ∫e^x dx = e^x + C | ∫e^(2x) dx = e^(2x)/2 + C |
| Trigonometric | ∫sin(x)dx = -cos(x) + C | ∫cos(x)dx = sin(x) + C |
Integration Techniques
- Substitution (u-substitution): Replace u = g(x) to simplify the integral
- Integration by Parts: ∫u dv = uv - ∫v du
- Partial Fractions: Decompose rational functions
- Trigonometric Substitution: Use trig identities to simplify
- Numerical Integration: Trapezoidal rule, Simpson's rule
Fundamental Theorem of Calculus
The fundamental theorem of calculus states that if F is continuous on [a,b] and F'(x) = f(x), then:
∫[a,b] f(x)dx = F(b) - F(a)
This connects derivatives and integrals, showing they are inverse operations.
Real-World Applications
- Physics: Finding displacement from velocity, work done by forces
- Engineering: Calculating areas, volumes, center of mass
- Economics: Computing total revenue from marginal revenue
- Medicine: Drug concentration over time, dose calculations
- Statistics: Probability distributions, cumulative functions
Practical Examples
∫x² dx = x³/3 + C
Check by differentiating: d/dx(x³/3) = x² ✓
∫[0,2] x² dx = [x³/3] from 0 to 2
= 8/3 - 0 = 8/3 ≈ 2.67
This is the area under x² from 0 to 2
∫(3x² + 2x) dx = ∫3x² dx + ∫2x dx
= x³ + x² + C
Explore More Tools
Calculus & Advanced Math
Basic Math & Algebra
Statistics & Analysis
Frequently Asked Questions
1. What's the difference between definite and indefinite integrals?
Indefinite integrals give a family of functions (with constant C). Definite integrals compute a specific numerical value representing area.
2. Why do we add +C to indefinite integrals?
The derivative of any constant is 0, so many functions have the same derivative. +C represents this family of antiderivatives.
3. What is the difference between ∫ and ∬?
∫ is a single integral (1D). ∬ is a double integral (2D, surface/area). ∭ is a triple integral (3D, volume).
4. How do I know which integration technique to use?
Look for patterns: power functions (power rule), products (integration by parts), fractions (partial fractions), compositions (substitution).
5. What are improper integrals?
Integrals with infinite limits or discontinuities. They may converge (have finite value) or diverge (infinite value).
6. Can every function be integrated?
Not symbolically. Some functions don't have closed-form antiderivatives, so numerical methods are used instead.
7. What's the relationship between integrals and areas?
Definite integrals compute signed area under a curve. Areas below the x-axis are negative, above are positive.
8. How do I check if my integral is correct?
Differentiate your result. If you get back the original function, your integral is correct.
9. What is a line integral?
Integration along a curve rather than over an interval. Used in physics for work (Force · distance along path).
10. What's the difference between Riemann sums and integrals?
Riemann sums approximate area with rectangles. Integrals are the limit of Riemann sums as rectangles get infinitely small.
11. How do substitution and chain rule relate?
u-substitution is the reverse of the chain rule. If F'(x) = f(g(x))g'(x), then ∫f(g(x))g'(x)dx = F(x) + C.
12. Is this calculator 100% accurate?
This calculator provides symbolic solutions for common integrals. Complex integrals may require numerical approximation.