∫ Integral Calculator
Find the indefinite integral (antiderivative) of any polynomial. Enter the coefficients from the highest degree to the constant term and get the integrated result instantly.
What is this tool?
Integration is the reverse process of differentiation. Given a function f(x), the indefinite integral ∫f(x)dx produces a new function F(x) whose derivative is f(x). Integration is one of the two pillars of calculus (alongside differentiation) and has applications across physics, engineering, economics and statistics. Areas under curves, total distance from velocity, accumulated growth, probability distributions and the work done by a variable force all involve integrals.
For polynomials, integration follows a simple power rule: ∫xⁿ dx = xⁿ⁺¹ / (n+1) + C, where C is the constant of integration. Each term is integrated independently. For example, ∫(3x² + 2x + 5) dx = x³ + x² + 5x + C. The constant C appears because the derivative of any constant is zero, so the original function could have had any constant added to it.
This integral calculator handles polynomials of any degree. You enter the coefficients from the highest power of x down to the constant term, and it applies the power rule to each term, showing the complete antiderivative. All computation happens locally in your browser.How it works
The power rule for integration states that for any term c·xⁿ (where n ≠ −1), the integral is c·xⁿ⁺¹ / (n+1) + C. The calculator processes each coefficient in order. If you enter coefficients [a, b, c, d] representing the polynomial ax³ + bx² + cx + d, the result is a·x⁴/4 + b·x³/3 + c·x²/2 + d·x + C.
The constant of integration C is always appended because indefinite integrals represent a family of functions, not a single function. In applied problems, C is determined by an initial condition or boundary value. The calculator displays the general antiderivative with C, and also simplifies fractional coefficients where possible (for example, 6x⁴/4 becomes 3x⁴/2).How to use
- Enter the coefficients from highest degree to constant, separated by commas.
- For example, enter 3,0,2,1 for 3x³ + 0x² + 2x + 1.
- Click Calculate to get the indefinite integral.
- Review the step-by-step integration of each term.
- Note the constant of integration C in the result.
Frequently Asked Questions
What is the constant of integration C?
The constant C represents the fact that infinitely many functions share the same derivative. Since the derivative of any constant is zero, adding a constant to an antiderivative produces another valid antiderivative. In real-world problems, C is determined by an initial condition (such as the position at time zero).
Can this calculator handle trigonometric or exponential functions?
No. This calculator uses the polynomial power rule, which applies only to terms of the form c·xⁿ. Trigonometric functions (∫sin x dx = −cos x + C), exponentials (∫e^x dx = e^x + C) and 1/x (∫(1/x) dx = ln|x| + C) require different rules that this tool does not cover.
What is the difference between indefinite and definite integrals?
An indefinite integral gives you a general function (the antiderivative) with a +C. A definite integral ∫[a,b] f(x)dx evaluates the antiderivative at two limits and subtracts, giving a number that represents the signed area under the curve between x=a and x=b. This calculator computes the indefinite integral.
Why does integrating x⁻¹ not work with the power rule?
The power rule gives xⁿ⁺¹/(n+1), but when n = −1, the denominator becomes zero, which is undefined. Instead, ∫(1/x) dx = ln|x| + C. This is a special case that requires logarithmic integration, which this polynomial-based calculator does not handle.
Tips & Advice
When integrating polynomials, keep track of each term separately and always add 1 to the exponent while dividing the coefficient by the new exponent. A common mistake is forgetting to divide: ∫x² dx is x³/3, not x³. Another common error is forgetting the constant of integration C in indefinite integrals. If you have a definite integral problem, first find the antiderivative using this tool, then substitute the upper and lower limits and subtract: F(b) − F(a). The constant C cancels out in definite integrals, so you do not need it there. For checking your work, differentiate the result: you should get back the original polynomial. If the derivative does not match, recheck each term. In physics, integration converts rates to totals: integrate velocity to get displacement, integrate power to get energy, integrate force (with respect to distance) to get work.
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