Integral Calculator
Calculate definite and indefinite integrals of common polynomial, exponential, trigonometric, and logarithmic functions with step-by-step solutions.
Calculate an integral
Select a function type and enter its values. For a definite integral, enter the lower and upper limits. Leave the limits empty to find an indefinite integral.
Your result
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| Integral | — |
| Limits | — |
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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:
First find the antiderivative:
∫3x² dx = x³
Evaluate the limits:
[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)
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Important: This calculator is provided for general informational and educational purposes only. The 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.