The antiderivative
Raise each power by one and divide by the new power. x² becomes x³/3, and a constant k becomes kx.
The + C is there because any constant differentiates to zero, so every antiderivative is one of a family.
Worked example
For x² − 3x + 1 the antiderivative is x³/3 − 3x²/2 + x + C.
Between −1 and 3 the definite integral is 1.333333, and the average value over that width of 4 is 0.333333.
Why the check matters
The page also evaluates the same integral numerically, using Simpson's rule with 1,000 strips.
When the two agree to nine decimal places, the algebra and the arithmetic have both been confirmed.
What it does not do
It integrates polynomials only. Integrals of sin, 1/x or exponentials need a computer algebra system.
Improper integrals, where a limit is infinite, are outside its range.
Inputs for the integral calculator
Coefficients
One box per power, from x⁴ down to the constant.
Lower and upper limits
The two ends of the definite integral. They can be in either order.
CALCULATOR QUESTIONS
Integral calculator questions
What is the integral of x² − 3x + 1?
x³/3 − 3x²/2 + x + C. Raise each power and divide by the new one.
What does the + C mean?
Any constant differentiates to zero, so the antiderivative is only fixed once you know one point.
How is the definite integral worked out?
Evaluate the antiderivative at the upper limit, subtract its value at the lower limit. For −1 to 3 that is 1.333333.
Why is the answer negative sometimes?
Area below the x-axis counts as negative, which is what a definite integral measures rather than plain area.