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Integral Calculator

Find antiderivatives and definite integrals for nine polynomial, power, trigonometric, exponential, logarithmic, radical, reciprocal, and constant families.

  • integration calculator
  • antiderivative calculator
  • definite integral calculator
  • trigonometric integral calculator
  • area under curve calculator

Use non-negative whole-number powers, such as 6x + 2 or 3x^2 − 4x + 1.

Starting x value for the definite integral.

Ending x value. Reversed bounds produce the negative integral.

Status: initial

Results

Awaiting calculation

Explore your result

Integral as accumulated change

Compare signed accumulation while keeping the selected family, antiderivative rule, bounds, and endpoint values visible.

Integral as accumulated change
Definite integral42

Use this result well

  1. 1Choose the matching function family.
  2. 2Check coefficients and bounds against the real domain.
  3. 3Review F(upper) and F(lower) before using their difference.

Reversing bounds changes the sign; a signed integral is not always the same as total geometric area.

Calculus guide

Integral calculator for nine function families

Choose a supported function family to find its symbolic antiderivative and evaluate a definite integral between two bounds. Conditional inputs keep each integration rule explicit. The guided selector returns both a symbolic antiderivative and the signed net accumulation between the supplied bounds for each supported function family.

When to use this integral calculator

Use it to check introductory calculus work involving polynomials, powers, reciprocal, sine, cosine, e-based exponential, natural logarithm, square root, and constant functions. It is a guided rule calculator, not an unrestricted computer algebra system. Use it to check introductory integration across the supported polynomial, power, trigonometric, exponential, logarithmic, radical, reciprocal, and constant families. It does not claim arbitrary symbolic substitution or advanced identities.

How integration is applied

The calculator builds an antiderivative F(x), includes the constant + C, then calculates F(upper) − F(lower). It validates real-number domains and rejects intervals containing singularities such as x = 0 for reciprocal functions. Each family applies its documented antiderivative rule. Every indefinite result includes + C, while the definite integral subtracts the lower antiderivative value from the upper value.

Variable explanations

Understand what each input and result means before calculating.

Function family

Selects the integration rule and relevant parameter fields.

Coefficient a

An outer multiplier carried through the antiderivative.

Exponent n

Power in a·xⁿ; exponent −1 uses the reciprocal logarithm rule.

Inner coefficient b

Multiplier in bx that becomes a divisor during integration.

Bounds

Lower and upper x values used in F(upper) − F(lower).

Reviewed by the Calculator.org.in Editorial Team

Formula behavior, validation cases, explanatory examples, and cited sources are checked before publication. This review supports educational accuracy and is not a substitute for qualified professional advice.

Last reviewed: 2026-08-23

Review process

Formula guide

See the calculation logic, variable definitions, and practical meaning.

Polynomial and power

∫axⁿ dx = a·xⁿ⁺¹/(n + 1) + C, n ≠ −1

  • a is coefficient.
  • n is exponent.

Apply the rule term by term for a polynomial.

Reciprocal

∫a/x dx = a ln|x| + C

  • The interval cannot contain x = 0.

The exponent −1 is the logarithmic exception to the normal power rule.

Sine

∫a sin(bx) dx = −(a/b) cos(bx) + C

  • b must be non-zero.
  • Angles use radians.

Divide by the inner coefficient and introduce a negative cosine.

Cosine

∫a cos(bx) dx = (a/b) sin(bx) + C

  • b must be non-zero.
  • Angles use radians.

Divide by the inner coefficient b.

Exponential

∫a e^(bx) dx = (a/b)e^(bx) + C

  • b must be non-zero.

The e-based exponential retains its form while dividing by b.

Natural logarithm

∫a ln(x) dx = a[x ln(x) − x] + C

  • Bounds must be greater than zero.

This identity follows from integration by parts.

Square root and constant

∫a√x dx = (2a/3)x³ᐟ² + C; ∫a dx = ax + C

  • Square-root bounds must be non-negative.

Square root is x¹ᐟ²; a constant gains a factor of x.

Worked examples

Follow realistic inputs through the calculation step by step.

1

Worked example

Polynomial

  1. 1Choose Polynomial and enter 6x + 2.
  2. 2The antiderivative is 3x^2 + 2x + C.
  3. 3From 0 to 2, the definite integral is 16.
2

Worked example

Reciprocal

  1. 1Choose Reciprocal with a = 2.
  2. 2Use bounds 1 and e.
  3. 3The definite integral is 2ln(e) − 2ln(1) = 2.
3

Worked example

Sine

  1. 1Choose Sine with a = 2 and b = 3.
  2. 2The antiderivative is −(2/3)cos(3x) + C.
  3. 3Evaluate it at each bound using radians.
4

Worked example

Natural logarithm

  1. 1Choose Natural logarithm with a = 1.
  2. 2Use bounds 1 and e.
  3. 3The definite integral equals 1.
5

Worked example

Find an indefinite polynomial integral

  1. 1Enter 3x^2 - 4x + 5.
  2. 2Integrate each term independently.
  3. 3The antiderivative is x³ - 2x² + 5x + C.
6

Worked example

Interpret reversed bounds

  1. 1Enter x and compare bounds 0 to 2 with bounds 2 to 0.
  2. 2The first definite integral is 2.
  3. 3Reversing the bounds changes only the sign, producing −2.

Common mistakes

Avoid these common input and interpretation errors.

Forgetting + C

An indefinite integral represents a family of antiderivatives differing by a constant.

Using the power rule at n = −1

The reciprocal case integrates to a logarithm instead.

Ignoring the inner coefficient

Sine, cosine, and exponential antiderivatives divide by b.

Crossing a singularity

A reciprocal or negative-power interval containing zero is improper and cannot use the ordinary endpoint formula.

Treating signed integral as total area

Regions below the x-axis subtract from the definite integral.

Frequently asked questions

Quick answers to the questions users ask most often.

Which integral functions are supported?
Polynomial, power, reciprocal, sine, cosine, e-based exponential, natural logarithm, square root, and constant functions.
Why is + C included?
All constants have derivative zero, so indefinite antiderivatives differ by an arbitrary constant.
Can the definite integral be negative?
Yes. It measures signed net accumulation, and reversing bounds changes the sign.
Are trigonometric bounds in degrees or radians?
Radians, as required by the standard calculus integration identities.
Why is a reciprocal interval containing zero rejected?
The function has a singularity at zero, making the interval an improper integral that cannot be evaluated by ordinary endpoint substitution.
Does this calculate total geometric area?
It calculates a signed definite integral. Total geometric area may require splitting at roots and adding absolute region areas.
Why is + C needed for an indefinite integral?
Every constant has derivative zero, so infinitely many antiderivatives differ only by a constant. + C represents that family.
Can a definite integral be negative?
Yes. A definite integral is signed net accumulation; regions below the x-axis contribute negatively, and reversing bounds changes the sign.
Is a definite integral always geometric area?
Not directly. It gives signed area. Total geometric area requires splitting at x-axis crossings and adding the absolute areas of the regions.

References

Sources used to support the calculator guidance.

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Version history

A transparent record of calculator content updates.

Updated 2026-08-23
  • 2.0.0 · 2026-08-23

    Expanded from polynomial-only integration to nine guided function families with conditional parameters, domain validation, symbolic antiderivatives, definite intervals, and endpoint steps.

  • 1.0.0 · 2026-08-03

    Initial polynomial antiderivative and definite integral calculator.