Integral Calculator
Calculate definite and indefinite integrals of common polynomial, exponential, trigonometric, and logarithmic functions with clear step-by-step solutions.
Calculate an integral
Choose the calculation mode, enter your function, and get the integral result with the applicable rule and working steps.
Your result
—
| Function | — |
| Integral | — |
| Limits | — |
| Result | — |
Step-by-step calculation
What is an integral?
An integral is a mathematical operation used to find accumulated quantities. A definite integral can represent the signed area between a function and the x-axis over an interval.
An indefinite integral represents a family of antiderivatives and therefore includes an arbitrary constant of integration.
How is an integral calculated?
Integration reverses differentiation. The calculator applies the appropriate antiderivative rule to the selected function.
For a definite integral, the resulting antiderivative is evaluated at the upper and lower limits and the lower value is subtracted from the upper value.
Integral calculator example
Consider the definite integral:
∫3x² dx = x³
[x³]02 = 2³ − 0³ = 8
Therefore, the value of the definite integral is 8.
Formula used
∫xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C, n ≠ −1
Constant multiple:
∫c f(x) dx = c ∫f(x) dx
Exponential:
∫eˣ dx = eˣ + C
Sine:
∫sin(x) dx = −cos(x) + C
Cosine:
∫cos(x) dx = sin(x) + C
Natural logarithm:
∫1/x dx = ln|x| + C
Definite integral:
∫ₐᵇ f(x) dx = F(b) − F(a)
Understanding Definite and Indefinite Integrals
Integration is one of the two fundamental operations of calculus. While differentiation describes how quickly a quantity changes, integration is commonly used to determine accumulated quantity from a rate or function. This makes integrals useful for finding areas, accumulated distance, total change, volume, and many other quantities represented by a mathematical function.
An indefinite integral gives a family of antiderivatives and therefore includes an arbitrary constant, written as C. A definite integral has a lower and upper limit and produces a numerical result after the antiderivative is evaluated at both endpoints.
Indefinite: ∫ f(x) dx = F(x) + C
How to Use the Integral Calculator
- Select Definite Integral or Indefinite Integral.
- Select the function type that matches your expression.
- Enter the coefficient, exponent, constant, or polynomial function.
- For a definite integral, enter both the lower and upper limits.
- Click Calculate Integral to display the result and calculation steps.
- Use Reset to start another calculation.
For a definite integral, both limits are required. For an indefinite integral, leave both limits empty.
Common Integration Rules
Power Rule
For a power of x, increase the exponent by one and divide by the new exponent, provided the original exponent is not −1.
Constant Multiple Rule
A constant coefficient can be taken outside the integral. This allows the remaining function to be integrated using its standard rule.
Exponential and Trigonometric Rules
The basic exponential and trigonometric functions have standard antiderivatives.
∫sin(x) dx = −cos(x) + C
∫cos(x) dx = sin(x) + C
Natural Logarithm
The reciprocal function 1/x has a logarithmic antiderivative.
What Does a Definite Integral Represent?
A definite integral represents the accumulated signed value of a function over an interval. When the function remains above the x-axis, the result corresponds to the area between the curve and the axis. If the function goes below the x-axis, that portion contributes negatively to the signed integral.
A definite integral is not always the same thing as total geometric area. If actual area is required for a function that crosses the x-axis, the interval may need to be split and the relevant areas treated separately.
Applications of Integration
Integration is used far beyond classroom calculus. It provides a mathematical way to accumulate small changes into a total quantity.
- Physics: velocity can be integrated to determine displacement, while acceleration can be integrated to obtain velocity.
- Engineering: integrals are used in signal analysis, mechanics, electrical systems, fluid flow, and structural calculations.
- Geometry: definite integrals can be used to calculate areas and volumes of regions described by functions.
- Economics: continuous rates and marginal quantities can be accumulated to estimate total values.
- Statistics: continuous probability distributions use integration to calculate probabilities over intervals.
How to Check an Integral Result
A useful way to verify an indefinite integral is to differentiate the antiderivative. The derivative should reproduce the original function. For a definite integral, evaluate the antiderivative at the upper and lower limits and subtract the lower value from the upper value.
Checking the sign of the answer is also useful. If a function is positive throughout an interval, its definite integral over that interval should not be negative.
Common Integration Mistakes
- Forgetting the + C in an indefinite integral.
- Using the power rule when the exponent is −1.
- Increasing the exponent without dividing by the new exponent.
- Using only one limit for a definite integral.
- Confusing a definite integral with total geometric area.
- Ignoring the domain of functions such as ln(x).
- Failing to verify a result by differentiating the antiderivative.
Accuracy and Calculator Limitations
This calculator is designed for common elementary integration problems and educational use. The supported function types are intentionally focused on standard forms that can be explained clearly with their corresponding integration rules.
More advanced integrals may require techniques such as substitution, integration by parts, partial fractions, numerical methods, or specialized symbolic mathematics. A numerical result should therefore be checked independently when precision is important.
For definite integrals, numerical evaluation can involve rounding. The displayed value should be treated as an approximation when the underlying calculation is evaluated numerically.
Frequently Asked Questions
What is an integral?
An integral is a calculus operation used to find an antiderivative or accumulate the value of a function over an interval. It is closely related to differentiation.
What is the difference between a definite and indefinite integral?
An indefinite integral produces an antiderivative plus an arbitrary constant C. A definite integral has lower and upper limits and produces a numerical value for the specified interval.
Why does an indefinite integral include + C?
Differentiating any constant gives zero. Therefore, many functions that differ only by a constant have the same derivative. The + C represents this family of possible antiderivatives.
What is the power rule for integration?
For n ≠ −1, the power rule is ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. The exponent is increased by one and the result is divided by the new exponent.
What happens when the exponent is −1?
The ordinary power rule does not apply because division by n + 1 would require division by zero. Instead, ∫(1/x) dx = ln|x| + C.
Can an integral be negative?
Yes. A definite integral is a signed accumulation. If a function is below the x-axis over part or all of an interval, that contribution is negative.
How can I verify an indefinite integral?
Differentiate the proposed antiderivative. If its derivative equals the original function, the antiderivative is correct up to an additive constant.
Why do I need both lower and upper limits?
A definite integral describes accumulation over a specific interval. Both endpoints are needed to define that interval.
Can this calculator solve every integral?
No. It is designed for the common function types shown in the calculator. More advanced integrals may require additional techniques or a symbolic mathematics system.
Related Calculators
Important: This calculator is provided for general informational and educational purposes only. Results are based on the function and values entered and standard integration rules. Numerical results are rounded for display and should be independently verified when high precision is required.