Integral Calculator
Integration is the reverse of differentiation. Where a derivative tells you the rate of change, an integral tells you the total accumulation.
Last updated: August 2026 · Written and verified by Akash Pandey
Integrate a polynomial
∫f(x)dx
This runs the same calculation engine as the main scientific calculator, entirely in your browser. Nothing you type is sent anywhere.
What is an integral?
An integral computes the area under a curve. The indefinite integral (antiderivative) gives you a family of functions; the definite integral gives you a number representing the total area.
The reverse power rule
For any term axⁿ, the indefinite integral is (a/(n+1))xⁿ⁺¹ + C. Add 1 to the exponent and divide by the new exponent - exactly the reverse of the derivative's power rule, which multiplies by the exponent and then subtracts 1. Integrating undoes differentiating, so the operations undo each other's two steps in reverse order.
∫axⁿ dx = (a/(n+1))xⁿ⁺¹ + C
- C is the constant of integration
- The reverse power rule works for all n ≠ −1.
Worked examples
Find the indefinite integral of f(x) = 6x²
∫6x² dx
- Apply the reverse power rule: increase the exponent from 2 to 3.
- Divide the coefficient by the new exponent:
6 ÷ 3 = 2. - Result:
2x³ + C.
= 2x³ + C
A car accelerates so that its velocity is v(t) = 2t² (m/s). How far does it travel in the first 3 seconds?
∫₀³ 2t² dt
- Given: a velocity function of time. Distance travelled is the accumulation of velocity, i.e. its definite integral.
- Method: find the antiderivative, then evaluate at the upper bound and subtract the value at the lower bound.
- Calculation: antiderivative is
(2/3)t³. Att = 3:(2/3)(27) = 18. Att = 0:0. - Interpretation:
18 − 0 = 18metres travelled - the definite integral turned a changing speed into one total distance.
= 18 metres
Real-world uses
- Distance from velocity - as above: integrating a speed over time accumulates it into a total distance, the mirror image of differentiating position to get speed.
- Area and volume - the definite integral is literally defined as the area between a curve and the x-axis, which is why it appears in any formula for the area of an irregular region or the volume of a solid built up from cross-sections.
- Total from a rate - integrating a rate of water flow gives total volume; integrating a production rate over time gives total output. Anywhere something accumulates from a changing rate, an integral is the right tool.
Frequently Asked Questions
What is the constant of integration, and why does it matter?
When you take an indefinite integral, any constant term present in the original function disappears during differentiation and cannot be recovered - "+ C" represents every constant it could have been. Dropping it is a common mistake because the rest of the answer looks identical without it.
What is a definite integral, and how is it different from an indefinite one?
An indefinite integral gives a family of functions (the antiderivative + C); a definite integral evaluates that antiderivative at two bounds and subtracts, collapsing the family into a single number - the net area under the curve between those points.
Can this calculator handle non-polynomial integrals?
Currently it handles polynomials using the reverse power rule. For trigonometric, exponential, or other integrals, use the main calculator.
What is the area under a curve, exactly?
The definite integral of a function between two points gives the signed area between the curve and the x-axis - area above the axis counts as positive, area below counts as negative, so the two can cancel.
What is the Fundamental Theorem of Calculus?
It links differentiation and integration: the integral of a derivative gives back the original function (up to a constant), and differentiating an integral returns the original function. It is the formal statement of why the two operations undo each other.
What happens when the region crosses the x-axis?
The definite integral still counts area above the x-axis as positive and below as negative, so a region straddling the axis can partially or fully cancel out. If you want total (unsigned) area instead, split the integral at the x-intercepts and take the absolute value of each piece before adding.
Why does the reverse power rule fail for n = −1?
The rule divides by n + 1, which is zero when n = −1 - division by zero is undefined. The integral of 1/x is not a power of x at all; it is ln|x| + C, which is why logarithms and integration are so closely linked.
How do I evaluate a definite integral with limits?
Find the antiderivative, then subtract its value at the lower limit from its value at the upper limit. For example, the integral of x² from 1 to 3 is (27/3) − (1/3) = 26/3 ≈ 8.667.