Integral Calculator
with steps — indefinite, definite, improper, double, triple & line integrals
Type any function and get the exact antiderivative with the rule used at every step — powers, 1/x, trig, exponentials, by parts and substitution. Switch tabs for definite, improper, double, triple and line integrals. Free, unlimited, no sign-up.
Online integral calculator — enter your function
Syntax: ^ power · * multiply · pi, e · functions sin cos tan sec csc cot ln log sqrt abs exp arctan arcsin. Example: 3x^2 + sin(2x) - 1/x.
How to Use the Integral Calculator
Four steps — the same flow for every type of integral.
Indefinite for the general antiderivative, Definite for area between limits, Improper for infinite limits, then Double, Triple or Line.
Use ^ for powers and * for multiply — or tap the keypad buttons. Bracket the inside of functions: sin(2x).
For definite and improper integrals enter a and b (you can use pi, e, inf, -inf).
Read the answer, then scroll to the step-by-step solution showing every rule used.
Integration of 1/x (and −1/x)
The power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) fails at n = −1 (division by zero), so 1/x needs its own rule.
- d/dx [ln x] = 1/x for x > 0
- d/dx [ln(−x)] = 1/x for x < 0
- Both cases combine into ln|x|
- ∫ 1/(ax + b) dx = (1/a)·ln|ax + b| + C
- ∫ k/x dx = k·ln|x| + C
- ∫₁ᵉ 1/x dx = ln e − ln 1 = 1
- ∫₀¹ 1/x dx diverges (ln 0 → −∞)
- ∫₁^∞ 1/x dx diverges (ln ∞ → ∞)
Worked Integral Examples
The same methods the calculator uses — handy for checking your own working.
∫ (3x² + 2x − 5) dx
- Integrate term by term: 3x³/3 + 2x²/2 − 5x
∫₀² (3x² + 2x) dx
- Antiderivative: x³ + x²
- F(2) − F(0) = (8 + 4) − 0
∫ x·eˣ dx
- u = x, dv = eˣ dx → du = dx, v = eˣ
- uv − ∫v du = x·eˣ − eˣ
∫ 2x/(x² + 1) dx
- Let u = x² + 1, so du = 2x dx
- ∫ du/u = ln|u|
∫₁^∞ 1/x² dx
- Antiderivative: −1/x
- Limit at ∞ is 0; at 1 it is −1 → 0 − (−1)
∬ xy dA over 0 ≤ x ≤ 1, 0 ≤ y ≤ 2
- Inner: ∫₀² xy dy = 2x
- Outer: ∫₀¹ 2x dx = 1
∭ (x + y + z) dV over the unit cube [0,1]³
- Each variable contributes ½ × 1 × 1
- Three terms: 3 × ½
∫C (x² + y²) ds on the unit circle x = cos t, y = sin t
- x² + y² = 1 and |r′(t)| = 1
- ∫₀^2π 1 dt
Common Integrals & Integration Rules
The standard results behind every step the calculator shows.
| Function f(x) | ∫ f(x) dx |
|---|---|
| k (constant) | kx + C |
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| aˣ | aˣ/ln a + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| tan x | −ln|cos x| + C |
| sec² x | tan x + C |
| csc² x | −cot x + C |
| 1/(1 + x²) | arctan x + C |
| 1/√(1 − x²) | arcsin x + C |
| ln x | x ln x − x + C |
Always add + C to an indefinite integral. For a definite integral the constant cancels: ∫ₐᵇ f dx = F(b) − F(a).
Types of Integral Calculators on This Page
Indefinite integral calculator
Finds the general antiderivative F(x) + C. Exact results for polynomials, 1/x, exponentials, trig functions, rational functions, and products solved by parts or substitution.
Definite integral calculator
Computes ∫ₐᵇ f(x) dx = F(b) − F(a) — the signed area under the curve. Limits can be numbers or expressions like pi/2.
Improper integral calculator
Handles infinite limits (inf, −inf) and endpoint singularities, and tells you whether the integral converges or diverges.
Double integral calculator online
Evaluates ∬ f(x,y) dA with constant limits or y-limits that depend on x — rectangles, triangles and regions between curves.
Triple integral calculator
Evaluates ∭ f(x,y,z) dV over a box or a region with variable y- and z-limits, integrating z first, then y, then x.
Line integral calculator
Computes ∫C f(x,y) ds along a curve given as x(t), y(t), using the arc-length element ds = √(x′² + y′²) dt.
Free Online Integral Calculator with Steps
This integral calculator is built for students who want more than a final number. For indefinite integrals it applies the real calculus rules — power rule, reciprocal rule, standard trig and exponential forms, partial fractions, integration by parts and u-substitution — and lists the rules it used, so you can copy the method into your answer sheet. Every symbolic answer is automatically checked by differentiating it numerically, so wrong antiderivatives are never shown.
As an online integral calculator it also covers the harder cases students search for: a definite integral calculator for area under a curve, an improper integral calculator for infinite limits, a double integral calculator and triple integral calculator that accept variable limits, and a line integral calculator for parametric curves. When a function has no elementary antiderivative (for example e^(−x²)), the definite and improper modes switch to accurate numerical integration and say so.
Everything runs in your browser, so results are instant and nothing you type is stored. Looking for related tools? Try the polynomial calculator, the GPA calculator or browse all free online calculators. For full explanations of limits, derivatives and integration, see our Maths homework help or ask the AI tutor below the calculator.
← Back to Maths Homework HelpIntegral calculator — common questions
The integral of 1/x is ln|x| + C. The absolute value is needed because 1/x is defined for negative numbers, while the natural log is only defined for positive ones. It follows that the integral of −1/x is −ln|x| + C.
Yes. For indefinite integrals it lists every rule applied (power rule, substitution, by parts, partial fractions and so on) and confirms the answer by differentiation. Definite, improper, double, triple and line integrals show the working for each stage.
An indefinite integral is the general antiderivative F(x) + C and is itself a function. A definite integral ∫ₐᵇ f(x) dx = F(b) − F(a) is a single number — the signed area under the curve between a and b — so no constant C is needed.
Enter inf or -inf as a limit in the Improper tab. The calculator evaluates the limit of the antiderivative (or uses a convergence-tested numerical method) and tells you whether the integral converges to a number or diverges. For example, ∫₁^∞ 1/x² dx = 1 but ∫₁^∞ 1/x dx diverges.
It integrates the inner variable first (y between its limits, which may depend on x) and then integrates that result over x. Enter y-limits such as 0 and x to integrate over a triangle, or constants for a rectangle.
For a scalar field f(x, y) along a curve x(t), y(t), the line integral is ∫ f(x(t), y(t))·√(x′(t)² + y′(t)²) dt over the chosen range of t. Enter the field, the parametric equations and the t-range and the calculator does the rest.
Some integrals (such as e^(−x²)) have no elementary antiderivative. In Definite and Improper mode the calculator then uses accurate numerical integration (Simpson's Rule) and labels the result as approximate. In Indefinite mode it tells you no standard rule applies.
Use ^ for powers (x^2), * for multiplication (3*x or just 3x), sqrt(x) for roots, and brackets for function inputs: sin(2x), ln(x), e^(3x), arctan(x). The keypad above inserts these for you. Use pi for π and inf for infinity.
Yes — completely free with no sign-up, no daily limit and no data stored. All calculations happen in your browser.