Integral Calculator — Definite Integrals, Step by Step
Type the coefficients of a polynomial up to x³, set your bounds, and get the exact definite integral — antiderivative by the power rule, then F(b) − F(a), every step shown.
Your integral
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Need sin, cos, exp, ln, roots, powers past x³, an indefinite integral, area between curves or a volume of revolution? Use the expression integrator below — same page, full steps.
Result
The area under the curve
The shaded region between your bounds is the integral — signed area above the x-axis minus area below it.
Integrate an expression — trig, exp, ln, roots and more
Type the integrand as you would write it: x^5, sin(x), exp(x), ln(x), sqrt(x), 1/x. This tool derives the antiderivative with the real rules — the power rule, substitution and integration by parts — and prints only the steps it actually performed. It also handles indefinite integrals, improper integrals with infinite bounds or a singularity inside the interval, the area between two curves, and volumes of revolution. Polynomials up to degree 12.
🔒 Parsed by a hand-written maths parser in your browser — never eval(), never sent anywhere.
Expression result
The rules this tool applies — nothing beyond this list is attempted, and an integrand outside the list gets an honest out-of-scope message, never a guessed number:
| Integrand shape | Rule applied | Antiderivative |
|---|---|---|
| x^n, any rational n ≠ −1 (roots included, e.g. x^(1/2)) | Power rule | x^(n+1) / (n + 1) |
| 1/x | Log rule | ln|x| |
| sin(x), cos(x) | Reversed derivatives | −cos(x), sin(x) |
| exp(x) | Reversed derivative | exp(x) |
| ln(x) | Integration by parts | x·ln(x) − x |
| sin(2x), exp(3x + 1), sqrt(2x + 1), 1/(2x + 1) | Substitution, u = ax + b | F(ax + b) / a |
| 2x·exp(x²), ln(x)/x, x/(x² + 1), x·sin(x²) | Substitution, u = the inside function | The antiderivative of the u-form, with u substituted back |
| x·exp(x), x·sin(x), x·cos(x), x·ln(x) | Integration by parts | u·v − ∫ v du, with u and dv stated in the steps |
| (x² + 3x + 1)/(x + 1) and similar | Polynomial division first | Quotient integrated term by term; a remainder over (ax + b) uses the log rule |
Not covered, by design: integrands needing rules outside this list — for example exp(x²), sin(x)/x, 1/(x² + 1) (an arctan form), tan and secant forms, or a product of two non-polynomial functions such as exp(x)·sin(x). Improper integrals are decided from the antiderivative’s limits: a finite limit gives the value, a limit that grows without bound is reported as divergence, and a case the rules cannot settle is reported as undecided rather than estimated.
- A definite integral is the antiderivative evaluated at the bounds: ∫ f(x) dx from a to b equals F(b) − F(a), with each term integrated by the power rule, ∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1).
- The worked default: the integral of 3x² from 0 to 2 is exactly 8 — F(x) = x³, so F(2) − F(0) = 8 − 0 = 8.
- The shaded region under the curve between your bounds is the integral — signed area above the x-axis minus area below it.
Keep exploring
Three companion tools for harder math.
How it works
Three steps from polynomial to exact answer.
Enter coefficients
Give a, b, c, and d for ax³ + bx² + cx + d — zeros drop terms you do not need.
Set the bounds
The lower and upper limits of integration — negative values and reversed order both work.
Apply F(b) − F(a)
The antiderivative is evaluated at both bounds and subtracted, with each step written out.
The math behind the integral
By the Fundamental Theorem of Calculus, a definite integral is the antiderivative evaluated at the bounds: ∫ab f(x) dx = F(b) − F(a). Each term uses the power rule: ∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1).
Worked examples:
| Integral | Antiderivative | Value |
|---|---|---|
| ∫₀² 3x² dx | F(x) = x³, so F(2) − F(0) = 8 − 0 | 8 |
| ∫₀¹ x³ dx | F(x) = x⁴/4, so F(1) − F(0) = 0.25 − 0 | 0.25 |
| ∫₁³ (2x + 1) dx | F(x) = x² + x, so F(3) − F(1) = 12 − 2 | 10 |
| ∫₋₁¹ (2x³ − 3x² + 5) dx | F(1) − F(−1) = 4.5 − (−3.5) | 8 |
Go deeper
The details that separate a rough guess from a trustworthy number.
Integration reverses differentiation, one term at a time:
- Cubic term: ∫ a·x³ dx = a·x⁴/4 — raise the power, divide by the new power.
- Quadratic term: ∫ b·x² dx = b·x³/3.
- Linear term: ∫ c·x dx = c·x²/2.
- Constant term: ∫ d dx = d·x — a constant integrates to a linear term.
The constant of integration (+C) cancels out in a definite integral, because F(b) − F(a) subtracts it away.
A definite integral measures net signed area, not total area:
- Above the axis counts positive: area between the curve and the x-axis where f(x) is positive.
- Below the axis counts negative: dips under the axis subtract from the total.
- Reversed bounds flip the sign: integrating from 2 to 0 gives the negative of integrating from 0 to 2.
- Symmetry shortcut: odd functions over symmetric bounds (like x³ from −1 to 1) integrate to exactly 0.
If you need total geometric area instead, the curve must be split wherever it crosses the axis — a different question from the signed integral.
Integration by parts: choose u so the new integral is easier
The formula ∫u dv = uv − ∫v du is a trade, and the trade only pays if ∫v du is easier than the integral you started with. The LIATE order — logarithmic, inverse trig, algebraic, trig, exponential — picks u correctly most of the time: for ∫ x eˣ dx, u = x gives ∫ x eˣ dx = eˣ(x − 1) + C in one step, while the swapped choice goes nowhere. When the new integral loops back to the original (as with eˣ sin x), that is not failure: collect the original on one side and solve for it algebraically.
Partial fractions: the algebra is the hard part
Partial fractions turn one hard rational integrand into several easy ones: factor the denominator, assign one term per factor (a linear numerator for irreducible quadratics, one term per power for repeated factors), solve for the constants by substituting convenient x-values, and integrate term by term to logarithms. The integration itself is routine; finding the constants is where care pays. Always check the decomposition by recombining — if the pieces do not add back to the original fraction, the constants are wrong, not the method.
Frequently asked questions
Quick answers about this integral calculator.
How does this calculator evaluate a definite integral?
The coefficient tool builds the antiderivative with the power rule — F(x) = a·x⁴/4 + b·x³/3 + c·x²/2 + d·x — then applies the Fundamental Theorem of Calculus: F(upper) − F(lower), shown step by step. The expression tool below derives the antiderivative with the power rule, substitution or integration by parts, then applies the same theorem.
What polynomials can I enter?
In the coefficient tool, any polynomial up to x³: ax³ + bx² + cx + d, with any real coefficients. In the expression tool below, polynomials up to degree 12 — plus sin, cos, exp, ln, square roots and 1/x, including functions of a linear expression such as sin(2x + 1).
Does this calculator use eval()?
No. The coefficient tool uses plain arithmetic on the numbers you enter. The expression tool reads what you type with a hand-written parser that only builds a maths tree — your input is never executed as code, and there is no eval() or Function() anywhere on this page.
What if the lower bound is bigger than the upper bound?
The sign flips: swapping the bounds negates the integral. The calculator still computes F(b) − F(a) in the order you entered, so integrating from 2 to 0 gives the negative of 0 to 2.
Why is my answer a long decimal?
Many exact answers are fractions — 1/3 shows as 0.33333333. Where the exact value is a rational or a standard form such as 11π/3 or e − 1, the expression tool shows the exact form with the decimal beside it; otherwise results display to 8 significant digits.
What does a negative result mean?
The integral is net signed area: between your bounds, more of the curve sits below the x-axis than above it. It is a meaningful answer, not an error.
How do I pick u for integration by parts?
Follow LIATE (logarithmic, inverse trig, algebraic, trig, exponential), choosing u as the highest-ranked type present. Verify the choice by checking that ∫v du is easier than the original.
When are partial fractions the right method?
For rational functions whose denominator factors. Split into simple fractions first; each then integrates to a logarithm or an arctangent.
What if polynomial division is needed first?
If the numerator degree is at least the denominator degree, divide first, integrate the quotient directly, and apply partial fractions to the remainder.
What do I do when parts loops back to the original integral?
Solve algebraically: move the original integral to one side and divide by its coefficient, as with ∫ eˣ sin x dx = (eˣ/2)(sin x − cos x) + C.
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Disclaimer
This calculator is for information and study only. The coefficient tool evaluates polynomials up to degree 3 exactly by the power rule. The expression tool on this page also derives antiderivatives for trig, exponential, logarithmic and root functions — using substitution and integration by parts — and handles improper integrals, areas between curves and volumes of revolution; when an integrand is outside the rules it applies, it says so instead of guessing. For graded coursework, always show your own working.