# Integral Calculator with Steps - Indefinite & Definite

> Step-by-step integral calculator: substitution, by parts, partial fractions, trig powers, completing the square. Every answer is checked by differentiation.

- URL: https://qilingtools.com/en/math/integral
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Type an integrand and get the antiderivative worked out the way a textbook does it. The solver splits sums and pulls out constants, then tries the table of standard integrals (with linear substitution), rational functions via polynomial long division and partial fractions (the denominator is factored into rational roots and quadratic factors, repeated factors included, with quadratics completed into arctan form), powers of trig functions (split off one factor for odd powers, half-angle identities for even powers, the tan substitution and the secⁿ reduction formula), u-substitution that recognises f(g(x))·g′(x) even up to a constant factor, completing the square under a square root (arcsin and ln forms), and integration by parts with u chosen by LIATE — including repeated parts such as x²eˣ and the cyclic case eˣ sin x where the original integral reappears and is solved for. Every step names the rule, shows the formula and the intermediate result. The final answer is then differentiated symbolically and compared with the integrand at several points, so you can see it is correct. Tick “definite integral” and enter limits to get both F(b) − F(a) and an adaptive Gauss–Kronrod numerical value, with a warning when a discontinuity makes them disagree. Integrals with no elementary antiderivative, such as e^(−x²) or sin(x)/x, are reported honestly and evaluated numerically instead. The plot shows the integrand, the antiderivative and the shaded area, and you can send both curves to the grapher.

## How to use

1. Type the integrand (e.g. x^2*e^x, 1/(x^2-1), sin(x)^3) and pick the variable if there are several letters
2. Read the antiderivative, the derivative check and the step-by-step solution
3. For a definite integral, tick the box and enter the limits to compare the exact and numerical values

## Features

- Substitution, parts (including cyclic), partial fractions, trig powers, completing the square
- Each step shows the rule, formula and intermediate result
- Answers are differentiated back and checked at several points
- Exact factoring of rational functions into linear and quadratic factors
- Definite integrals: F(b) − F(a) cross-checked with Gauss–Kronrod quadrature
- Honest “no elementary antiderivative” message with a numerical fallback
- Plot of f, F and the area, with a link to the grapher
- Copy the result as pretty text or plain expression

## FAQ

### Why does my answer look different from the calculator’s?

Antiderivatives are only defined up to a constant, and the same function can be written in many ways: ∫sin x cos x dx is sin²x/2, but also −cos²x/2 or −cos 2x/4. The tool differentiates its answer and compares it with your integrand; “F′(x) matches f(x) at n points” means the result is right. To compare with your own answer, subtract the two and check that the difference is a constant.

### Which integrals can’t be done?

Integrals such as e^(−x²), sin(x)/x, √(1 + x³) and xˣ have no antiderivative in elementary functions, so no method can produce one; the tool says “no elementary step-by-step solution found” and gives the numerical definite integral instead. It also gives up on partial fractions when the denominator can’t be split into linear and quadratic factors with rational coefficients (1/(x⁴ + 1) needs √2), again falling back to numerical integration.

### Why do the two definite-integral values disagree?

The fundamental theorem F(b) − F(a) only applies when the integrand is continuous on [a, b]. If there’s a discontinuity or singularity inside, as in ∫ from −1 to 1 of 1/x² dx, plugging into the formula gives a wrong finite number, while the numerical integration warns that it doesn’t converge. Both values are shown, and when they differ you should check for discontinuities and treat it as an improper integral.

### How is u chosen in integration by parts?

By LIATE: logarithmic > inverse trig > algebraic (polynomials and powers) > trig > exponential; the first one present becomes u and the rest is dv. When ln x or arctan x appears alone, dv = dx. A polynomial times an exponential or trig function is integrated by parts repeatedly until the polynomial is a constant. For eˣ times a trig function the original integral reappears after two rounds, and the solver moves it to the left side — the step “I = … − I” is shown.

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