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Integral Calculator

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Enter your polynomial's coefficients to find its indefinite integral instantly, using the power rule.

Integral Calculator

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f(x) = 3x² + 5x - 2
∫f(x)dx
x³ + 2.5x² - 2x + C
Covers polynomials (terms with x raised to a whole-number power) - not trigonometric, logarithmic, or other non-polynomial functions.

The formula (power rule for integration)

∫(axⁿ)dx = (a / (n+1)) × xⁿ⁺¹ + C Applied to each term, then combined
Example

f(x) = 3x² + 5x - 2: integral of 3x² is x³, of 5x is 2.5x², of -2 is -2x. Result: x³ + 2.5x² - 2x + C.

Step-by-step guide

  1. Enter the coefficient for each power of x in your polynomial, leaving unused terms at 0.
  2. Read the indefinite integral, updated instantly as you type.

Why every indefinite integral includes "+ C"

The derivative of any constant is zero, which means going in reverse - integrating - can never fully recover what constant, if any, was originally part of the function. If f(x) = x³ + 5 and g(x) = x³ + 12, both have the exact same derivative (3x²), so integrating 3x² genuinely cannot distinguish between them. The "+ C" explicitly represents this entire family of possible original functions, all differing only by a constant - it's not a formality, it's a mathematically necessary acknowledgment of information that integration alone cannot recover.

Common mistakes

Forgetting the "+ C" - the indefinite integral of a function is a family of functions, not a single one, and dropping the constant loses that distinction.
Applying the polynomial power rule to non-polynomial terms (like 1/x or eⁿ) - those follow entirely different integration rules not covered here.

Frequently asked questions

What's the difference between this and a definite integral?

This calculates the indefinite integral (a general antiderivative expression with "+ C"). A definite integral instead evaluates that antiderivative between two specific bounds, producing a single number rather than an expression.

Does this handle 1/x (which would need ln|x|)?

No - the power rule shown here breaks down specifically at x⁻¹ (1/x), which integrates to ln|x| instead of following the standard pattern. This calculator is built for standard polynomial terms only.

Can I find the exact value of C?

Not from the integral alone - you need an additional piece of information, like a known point the original function passes through, to solve for the specific value of C.

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