What Is an Integral Calculator?
An integral calculator finds the antiderivative of a polynomial function and shows you each step. You type in a function like 3x^2 + 2x - 5, and the tool applies the power rule for integration term by term, adds 1 to each exponent, divides by the new exponent, and appends the constant of integration C. You can also calculate a definite integral by entering lower and upper bounds, which gives you the net area under the curve between those two points.
Integration is the inverse operation of differentiation. If you differentiate a function and then integrate the result, you get back the original function (plus a constant). This relationship is called the Fundamental Theorem of Calculus, and it is one of the most important results in mathematics. Integration was developed alongside differentiation by Newton and Leibniz in the late 1600s. The integral symbol (the long S) was introduced by Leibniz to suggest a continuous sum.
How Integration Works
There are two types of integrals. An indefinite integral produces a family of functions (the antiderivative plus a constant C). A definite integral produces a number representing the signed area between the function and the x-axis over a specified interval.
The Power Rule for Integration
The power rule for integration is the reverse of the power rule for differentiation:
[a x^n] dx = (a / (n+1)) x x^(n+1) + C
You add 1 to the exponent, then divide the coefficient by the new exponent. For example, the integral of 3x^2 is (3/3) x x^3 = x^3 + C. The integral of 4x is (4/2) x x^2 = 2x^2 + C. The integral of 5 (a constant) is 5x + C, because 5 = 5x^0, so (5/1) x x^1 = 5x.
The Constant of Integration (C)
Every indefinite integral ends with + C. This is because the derivative of any constant is zero, so when you reverse the process, you cannot know what constant was originally there. The derivative of x^3 + 7 is 3x^2, and so is the derivative of x^3 - 42. Both have the same derivative. So the integral of 3x^2 could be x^3 + 7 or x^3 - 42 or x^3 + any other number. We write x^3 + C to represent all possibilities.
Definite Integrals
A definite integral calculates the net signed area between a function and the x-axis from a lower bound a to an upper bound b. You evaluate the antiderivative at both bounds and subtract: F(b) - F(a). The constant C cancels out in this subtraction, which is why definite integrals produce a specific number rather than a family of functions. If the function is above the x-axis, the area is positive. If it is below, the area is negative.
How to Use the Calculator
- Type your polynomial function in the input field. Use x as the variable and ^ for exponents.
- For an indefinite integral, leave the bounds empty. The result will include + C.
- For a definite integral, enter both lower and upper bounds. The calculator evaluates the antiderivative at both points and subtracts.
- The integral and step-by-step solution appear instantly on the right.
Example Calculations
Example 1: Indefinite Integral
Find the integral of f(x) = 3x^2 + 2x - 5:
- [3x^2] dx: (3/3) x x^3 = x^3
- [2x] dx: (2/2) x x^2 = x^2
- [-5] dx: (-5/1) x x^1 = -5x
- Result: F(x) = x^3 + x^2 - 5x + C
Example 2: Definite Integral
Find the definite integral of f(x) = 6x^2 + 4x + 1 from x = 0 to x = 3:
- Antiderivative: F(x) = 2x^3 + 2x^2 + x + C
- F(3): 2(27) + 2(9) + 3 = 54 + 18 + 3 = 75
- F(0): 0 + 0 + 0 = 0
- Definite integral: 75 - 0 = 75
- Interpretation: The area under 6x^2 + 4x + 1 from 0 to 3 is 75 square units.
Real-World Scenarios
Calculating Distance from Velocity
A physics student knows that velocity is the derivative of position, so position is the integral of velocity. If a car's velocity is v(t) = 6t^2 + 4t + 1 meters per second, the position function is s(t) = 2t^3 + 2t^2 + t + C. To find the distance traveled in the first 3 seconds, the student calculates the definite integral from 0 to 3, which gives 75 meters. This is a direct application of the Fundamental Theorem of Calculus. For finding the rate of change in the opposite direction, use our Derivative Calculator.
Computing Total Cost from Marginal Cost
An economist knows that the marginal cost function MC(x) = 0.06x^2 + 2x + 5 represents the cost of producing one additional unit at production level x. The total cost function is the integral of marginal cost: TC(x) = 0.02x^3 + x^2 + 5x + C, where C is the fixed cost. If the fixed cost is 1,000 dollars, then TC(x) = 0.02x^3 + x^2 + 5x + 1000. At x = 50 units, the total cost is TC(50) = 0.02(125000) + 2500 + 250 + 1000 = 2500 + 2500 + 250 + 1000 = 6250 dollars.
Finding Area Under a Curve
A surveyor needs to find the area of a plot of land bounded by a curved edge modeled by f(x) = 2x^4 - x^2 + 3 and the x-axis from x = 0 to x = 2. The antiderivative is F(x) = (2/5)x^5 - (1/3)x^3 + 3x. Evaluating: F(2) = (2/5)(32) - (1/3)(8) + 6 = 12.8 - 2.667 + 6 = 16.133. F(0) = 0. The area is approximately 16.13 square units. This is how engineers and surveyors use integration to compute areas of irregular shapes. For solving the polynomial equations that arise in these calculations, see our Quadratic Equation Calculator.
Common Mistakes to Avoid
- Forgetting the constant of integration: Every indefinite integral must include + C. Without it, the answer is incomplete. The constant matters because the derivative of x^3 + 5 is the same as the derivative of x^3 + 100. Integration cannot recover the original constant.
- Dividing by zero when n = -1: The power rule for integration requires dividing by n+1. If n = -1, you would divide by zero. The integral of x^(-1) = 1/x is ln|x| + C, not a power rule application. This calculator handles polynomials with non-negative exponents.
- Subtracting in the wrong order for definite integrals: The definite integral is F(upper) - F(lower), not F(lower) - F(upper). Reversing the order gives the negative of the correct answer. Always evaluate at the upper bound first and subtract the lower bound evaluation.
- Confusing integration with differentiation: Integration increases the power by 1 and divides by the new power. Differentiation decreases the power by 1 and multiplies by the old power. The integral of 3x^2 is x^3 + C, but the derivative of 3x^2 is 6x.
- Not simplifying fractions: The integral of 4x^3 is (4/4)x^4 = x^4, not (4/4)x^4. Always simplify the coefficient fraction. The integral of 6x^2 is (6/3)x^3 = 2x^3, not (6/3)x^3.
Authoritative Research and Resources
- Khan Academy: Integrals - A free course covering the Fundamental Theorem of Calculus, indefinite and definite integrals, the power rule for integration, and applications. Includes video lessons and practice problems with immediate feedback.
- OpenStax Calculus: Integration Chapter - A peer-reviewed, open-source calculus textbook covering integration rules, the Fundamental Theorem of Calculus, and applications to area, volume, and average value. Used in university courses worldwide.
- Paul's Online Math Notes: Indefinite Integrals - Paul Dawkins' widely used calculus reference from Lamar University. Provides clear explanations of integration rules with worked examples and practice problems, including the power rule and common integration formulas.