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Integral Calculator

Calculate the indefinite integral (antiderivative) of polynomial terms using the reverse power rule. Enter a coefficient and exponent to find the antiderivative, a core operation in calculus for computing areas and accumulated quantities.

Reviewed by Christopher FloiedPublished Updated

This free online integral calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

The coefficient of the term ax^n

The exponent of x in ax^n (must not be -1)

Results

New Coefficient

2

New Exponent

3

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Integral Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Integral Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Integral Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Integral Calculator

The Integral Calculator computes the indefinite integral (antiderivative) of monomial terms of the form ax^n using the reverse power rule. Integration is the inverse process of differentiation and one of the two central operations in calculus. Finding antiderivatives allows you to compute areas under curves, total displacement from velocity functions, accumulated revenue from marginal revenue, and volumes of solids of revolution. This calculator handles the mechanical computation so students and professionals can focus on setting up the problem correctly and interpreting the result. Remember that every indefinite integral includes an arbitrary constant C.

The Math Behind It

Integration reverses differentiation. If the derivative of F(x) is f(x), then F(x) is an antiderivative of f(x), and the indefinite integral of f(x)dx equals F(x) + C, where C is an arbitrary constant. The reverse power rule states: the integral of ax^n dx = [a/(n+1)]x^(n+1) + C, provided n is not equal to -1 (when n = -1, the integral is a*ln|x| + C). The Fundamental Theorem of Calculus, proved by Newton and Leibniz, connects differentiation and integration. It states that if F is an antiderivative of f on [a,b], then the definite integral from a to b of f(x)dx equals F(b) - F(a). This theorem transforms the problem of computing areas into finding antiderivatives. Historically, Archimedes computed areas under parabolas using the method of exhaustion, essentially an early form of integration. The formal development of integral calculus in the 17th century by Newton and Leibniz unified these techniques under a single framework. Bernhard Riemann later provided the rigorous definition of the integral as a limit of Riemann sums. Integration has vast applications: in physics, integrating force over distance gives work; integrating charge density gives total charge. In probability, integrating a probability density function over an interval gives the probability of the variable falling in that interval. In engineering, integration is used in signal processing, control theory, and structural analysis. The reverse power rule handles polynomial integrands, which appear frequently in approximations via Taylor series.

Formula Reference

Reverse Power Rule

integral of ax^n dx = [a/(n+1)] * x^(n+1) + C

Variables: a = coefficient, n = exponent (n != -1), C = constant of integration

Worked Examples

Example 1: Integral of 6x^2

Find the indefinite integral of f(x) = 6x^2.

Step 1:Identify a = 6, n = 2
Step 2:Apply the reverse power rule: new coefficient = 6 / (2+1) = 6/3 = 2
Step 3:New exponent = 2 + 1 = 3
Step 4:Write the result: integral = 2x^3 + C

The integral is 2x^3 + C.

Example 2: Integral of 4x^3

Find the indefinite integral of f(x) = 4x^3.

Step 1:Identify a = 4, n = 3
Step 2:New coefficient = 4 / (3+1) = 4/4 = 1
Step 3:New exponent = 3 + 1 = 4
Step 4:Result: integral = x^4 + C

The integral is x^4 + C.

Common Mistakes & Tips

  • !Forgetting to add the constant of integration C. Every indefinite integral has an arbitrary constant because the derivative of a constant is zero.
  • !Using the reverse power rule when n = -1. The integral of x^(-1) is ln|x| + C, not x^0/0.
  • !Adding 1 to the coefficient instead of dividing. The reverse power rule divides the coefficient by (n+1), it does not add 1 to it.
  • !Forgetting to increase the exponent by 1. Both operations (divide coefficient, increase exponent) must be performed together.

Related Concepts

Derivative

The inverse operation of integration. Taking the derivative of an antiderivative returns the original function.

Area Under Curve

Uses definite integration to compute the exact area between a function and the x-axis over an interval.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Area Under Curve Calculator

Derivative Calculator

Logarithm Calculator

Frequently Asked Questions

What is an indefinite integral?

An indefinite integral finds the general antiderivative of a function. It represents a family of functions differing only by a constant C. For example, the integral of 2x dx is x^2 + C, because the derivative of x^2 + C is 2x for any constant C.

Why do we add + C to indefinite integrals?

The constant C accounts for the fact that many different functions can have the same derivative. For instance, x^2, x^2 + 5, and x^2 - 3 all have derivative 2x. The constant C represents this family of solutions.

What happens when the exponent is -1?

When n = -1, the reverse power rule formula produces division by zero. The integral of x^(-1) = 1/x is the natural logarithm: ln|x| + C. This is a special case that must be memorized separately.

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