Integral Calculator
Calculate definite and indefinite integrals with step-by-step solutions
Use: x^2, sqrt(x), sin(x), cos(x), tan(x), ln(x), e^x, pi
More intervals = higher accuracy but slower calculation
∫ f(x) dx ∫ f(x) dx
Area under the curve Antiderivative of f(x)
Calculation Details
Power Functions
∫xⁿ dx = xⁿ⁺¹/(n+1) + C
∫1/x dx = ln|x| + C
∫√x dx = (2/3)x^(3/2) + C
Trigonometric
∫sin(x) dx = -cos(x) + C
∫cos(x) dx = sin(x) + C
∫sec²(x) dx = tan(x) + C
Exponential & Log
∫eˣ dx = eˣ + C
∫aˣ dx = aˣ/ln(a) + C
∫ln(x) dx = x·ln(x) - x + C
Integration Rules
∫k·f(x) dx = k·∫f(x) dx
∫[f+g] dx = ∫f dx + ∫g dx
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About Integral Calculator
What is an Integral?
An integral is a fundamental concept in calculus that represents the area under a curve or the accumulation of quantities. Integration is the inverse operation of differentiation and is essential for solving problems in physics, engineering, economics, and many other fields.
How to Use This Calculator
- Select Integration Type: Choose between definite (numerical) or indefinite (symbolic) integration
- Enter Function: Type your function using standard mathematical notation
- Set Bounds (for definite): Enter the lower and upper limits of integration
- Calculate: Click calculate to get your result
Types of Integrals
Definite Integrals
A definite integral calculates the exact area under a curve between two points (bounds). The result is a numerical value.
Formula: ∫ₐᵇ f(x) dx
Indefinite Integrals
An indefinite integral finds the antiderivative of a function. The result is a function plus a constant of integration (C).
Formula: ∫ f(x) dx = F(x) + C
Numerical Method: Simpson's Rule
This calculator uses Simpson's Rule for numerical integration, which provides excellent accuracy:
Formula: ∫ₐᵇ f(x) dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)]
Where h = (b-a)/n and n is the number of intervals.
Common Integral Formulas
| Function | Integral |
|---|---|
| xⁿ | xⁿ⁺¹/(n+1) + C |
| 1/x | ln |
| eˣ | eˣ + C |
| sin(x) | -cos(x) + C |
| cos(x) | sin(x) + C |
| sec²(x) | tan(x) + C |
Frequently Asked Questions
What is the difference between definite and indefinite integrals?
Definite integrals have bounds and give a numerical value (area). Indefinite integrals have no bounds and give a family of functions (antiderivatives).
How accurate is Simpson's Rule?
Simpson's Rule is very accurate for smooth functions. This calculator uses 1000 intervals, providing precision to at least 6 decimal places for most functions.
What functions are supported?
This calculator supports polynomials, trigonometric functions (sin, cos, tan), exponential functions (e^x, exp), logarithms (ln, log), square roots (sqrt), and combinations of these.
When should I use numerical vs symbolic integration?
Use numerical (definite) when you need an exact area or value. Use symbolic (indefinite) when you need the general antiderivative formula.
Integration Rules
Power Rule
∫xⁿ dx = xⁿ⁺¹/(n+1) + C, where n ≠ -1
Constant Multiple Rule
∫k·f(x) dx = k·∫f(x) dx
Sum Rule
∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
Integration by Substitution
For composite functions, use u-substitution
Note: This calculator provides numerical approximations for definite integrals and symbolic results for common functions. For complex integrals requiring advanced techniques (integration by parts, partial fractions), consult a mathematics textbook or professional.