Antiderivative Calculator

1. Enter your polynomial using standard math notation. Use ^ for exponents (so x^2 means x squared) and + / - between terms.
2. The calculator integrates instantly, applying the power rule to each term and adding the constant of integration C.
3. Read the step-by-step breakdown below the result — it shows the rule applied to each individual term and the resulting antiderivative for that term.
4. For non-polynomial functions (sin, cos, e^x, ln, etc.), use the reference table at the bottom to look up the antiderivative manually.
5. For definite integrals, compute the antiderivative F(x), then evaluate F(b) − F(a) at the bounds.

The power rule is the most-used rule in introductory calculus for finding antiderivatives of polynomial terms:

∫ x^n dx = x^(n+1) / (n+1) + C   (for n ≠ −1)
∫ 1/x dx = ln|x| + C   (special case for n = −1)

The general approach for any polynomial: integrate term by term, applying the power rule and pulling out constants. The constants of integration combine into a single + C.

Example: ∫ (3x² + 2x + 5) dx

∫ 3x² dx = 3 · x³/3 = x³
∫ 2x dx  = 2 · x²/2 = x²
∫ 5 dx   = 5x
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Answer: x³ + x² + 5x + C

Verify by differentiating: d/dx(x³ + x² + 5x + C) = 3x² + 2x + 5 ✓

These three terms are often used interchangeably, but they have subtly different meanings worth understanding before your next calculus exam.

The Fundamental Theorem of Calculus ties these together: to compute a definite integral, you find any antiderivative F(x) and evaluate at the bounds. The + C cancels out, which is why definite integrals produce specific numerical answers rather than functions.

Beyond the power rule, three integration techniques cover almost all of single-variable calculus:

• U-substitution — the reverse of the chain rule. Use when you see a function and its derivative both present.
• Integration by parts — the reverse of the product rule: ∫ u dv = uv − ∫ v du. Use for products of unlike functions (like x·sin(x) or x·e^x).
• Partial fractions — for rational functions where the denominator factors. Decompose into simpler fractions before integrating.