Find the antiderivative of a polynomial function and calculate its definite integral between two bounds.
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The integral calculator finds the "antiderivative" of a polynomial function using the reverse power rule for integration: the integral of the term axⁿ equals (a/(n+1)) xⁿ⁺¹, meaning the exponent is increased by one and then the coefficient is divided by the new exponent, plus the constant of integration C, which is always added because any constant number disappears upon differentiation. If a range is specified, the calculator also calculates the "definite integral" by substituting the upper and lower bounds into the antiderivative and subtracting the two results, which represents the area enclosed between the function's curve and the x-axis between points a and b. Integration is widely used in calculating irregular areas and volumes, and in physics to calculate total displacement from a velocity function. The constant of integration C matters specifically for an indefinite integral, since infinitely many functions share the exact same derivative, differing only by whatever constant value was originally attached to them.
Integration and differentiation are inverse operations of each other, in precisely the same sense that addition undoes subtraction. Where differentiation finds a function's instantaneous rate of change, integration works backward from that rate of change to reconstruct the original function — which is exactly why finding an integral is also called finding an 'antiderivative'.
The reverse power rule follows directly from undoing the ordinary power rule step by step: since differentiating axⁿ multiplies by n and reduces the exponent by one, integrating must do the opposite — increase the exponent by one, then divide by that new exponent, restoring the coefficient to what it would have needed to be before differentiation removed it.
The constant of integration, C, exists because differentiation permanently erases any constant term from a function — the derivative of x² and the derivative of x² + 7 are identical, since a constant's rate of change is always zero. This means that when reversing the process, there is no way to know from the derivative alone what constant, if any, the original function contained, so a general placeholder C must always be included in an indefinite integral to represent every possible original function consistent with that derivative.
A definite integral resolves this ambiguity by evaluating the antiderivative at two specific bounds and subtracting the results — the constant C cancels out completely in this subtraction, since it is added at both bounds and then subtracted away. What remains has a direct geometric meaning: the exact area enclosed between the function's curve and the x-axis, between the two chosen bounds.
This area-under-a-curve interpretation is why integration is the standard tool for measuring anything irregular that cannot be captured by a simple geometric formula — the area of a shape bounded by a curved edge, the volume of an irregularly shaped solid, or the total accumulated effect of a quantity that changes continuously over time or space.
Physics uses this accumulation property directly: if velocity is known as a function of time, integrating it over a time interval gives the total displacement during that interval, since displacement is exactly the accumulated effect of velocity over time — the reverse of how differentiating position gives velocity in the first place. Engineering, economics, and probability all rely on the same core idea whenever a total quantity needs to be reconstructed from a known rate of change.
Because differentiating any constant gives zero, so infinitely many antiderivatives differ only by a constant — C represents that unknown constant.
It represents the net signed area between the function's curve and the x-axis, between the two bounds you specify.
Yes, leave the bounds fields empty and the calculator will only show the general antiderivative with '+ C'.