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Integral Calculator

Enter the mathematical function you want to integrate. For definite integrals, provide the upper and lower bounds.

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Indefinite

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Upper limit:

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Integral Calculator

Use this free online Integral Calculator to evaluate integrals of any mathematical function easily. It supports both indefinite integrals (to find antiderivatives) and definite integrals (to compute numerical values over a specific interval).

What Is an Integral in Calculus?

In calculus, an integral represents the accumulation of quantities like area, volume, displacement, or total change by summing infinitely small contributions. Integration, the reverse of differentiation, is essential in mathematics, physics, and engineering, especially when analyzing continuous change or irregular shapes.

The process of computing an integral is called integration, and the function being integrated is called the integrand.

For example:

∫ 3x dx

∫ – integral symbol
3x – integrand
dx – integration with respect to x

For a definite integral, the result gives the area under the curve between two limits, written as A = ∫ₐᵇ f(x) dx.

Definite vs Indefinite Integrals

Indefinite Integrals:

Provide the antiderivative of a function without limits. Includes a constant of integration (+C).

Definite Integrals:

Computed over a fixed interval [a, b]. Evaluated as the limit of a sum as the partition width approaches zero, yielding a numerical value.

Integral Example

Basic Integration Formulas

Power Rule

∫ 1 dx = x + C
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, n ≠ -1

Constant Rule

∫ a dx = ax + C

Reciprocal Rule

∫ (1/x) dx = ln|x| + C

Exponential & Logarithmic Rules

∫ aˣ dx = aˣ / ln|a| + C, a > 0, a ≠ 1
∫ eˣ dx = eˣ + C

Trigonometric Integrals

∫ sin(x) dx = -cos(x) + C
∫ cos(x) dx = sin(x) + C
∫ tan(x) dx = -ln|cos(x)| + C
∫ cosec²(x) dx = -cot(x) + C
∫ sec²(x) dx = tan(x) + C
∫ cot(x) dx = ln|sin(x)| + C
∫ sec(x)tan(x) dx = sec(x) + C
∫ cosec(x)cot(x) dx = -cosec(x) + C

Inverse Trigonometric Integrals

∫ 1/√(1 - x²) dx = sin⁻¹(x) + C
∫ 1/(1 + x²) dx = tan⁻¹(x) + C
∫ 1/(|x|√(x² - 1)) dx = sec⁻¹(x) + C

Instead of memorizing formulas, simply enter your function into our integral solver for quick and accurate solutions.

How to Solve Integrals Manually

Indefinite Integrals:

Identify the integrand f(x)
Apply appropriate integration rules
Add the constant of integration (+C)

Example: ∫ (x³ + 5x + 6) dx

∫ x³ dx = x⁴ / 4
∫ 5x dx = 5x² / 2
∫ 6 dx = 6x

Final Answer: x⁴/4 + 5x²/2 + 6x + C

Definite Integrals:

Identify f(x) and limits [a, b]
Find antiderivative F(x)
Compute F(b) - F(a)

Example: ∫ sin(x) dx from 0 to π/2

F(x) = -cos(x)
F(π/2) = 0
F(0) = -1
Result = 1

How to Use the Integral Calculator

Enter the function
Select the variable
Choose definite or indefinite option
Enter bounds if definite
Click "Calculate"

FAQs

Is an Antiderivative the Same as an Integral?

An antiderivative is a single function whose derivative equals the integrand, whereas an indefinite integral represents all such functions.

Applications of Integration

Physics (motion, work, energy)
Fluid dynamics
Signal processing
Heat transfer

Why Include the Constant of Integration?

The constant (+C) represents all possible original functions since the derivative of a constant is zero.

How Does a Definite Integral Represent Area?

It calculates the net area between the curve and the x-axis over the given interval.

References

Wikipedia: Integral

Math Is Fun: Definite Integration

Math.com: Integration Formulas

Toppr: Integration in Daily Life

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EN	TR	ID
PL	DE	JA
KO	CS	PT
FR	ES	IT
RU	AR	FI
DA	NO	SL
TH	NL