What kinds of integrals can this calculator solve?

It solves indefinite and definite integrals of polynomials, rational functions of the form 1/(ax+b), trigonometric functions (sin, cos, tan, sec, csc, cot), hyperbolic functions, exponentials such as eˣ and aˣ, the natural logarithm ln(x), and any of the above with a linear inner argument (u-substitution with u = ax+b).

How do I enter a definite integral?

Open the Bounds panel under the keypad and type the lower and upper limits (numbers, π, e or simple expressions). The calculator will compute the antiderivative F(x) and then F(upper) − F(lower) automatically.

What does + C mean in the answer?

An indefinite integral has infinitely many antiderivatives that differ by a constant. The + C represents this arbitrary constant of integration. For definite integrals the constant cancels when you subtract F(lower) from F(upper).

Why does the calculator say a problem is not supported?

Some integrals require advanced techniques such as integration by parts on products like x·sin(x), trigonometric substitution, partial fractions for non-linear denominators, or series expansion. These are intentionally outside the scope of a step-by-step student tool. Try splitting the integrand into simpler terms or rewriting it before integrating.

∫ Calculus Integral Calculator

Compute indefinite and definite integrals step-by-step. Type the integrand directly or use the specialised keyboard for trig, log, exponential and power functions.

Enter the integrand (use x as the variable)

x^2 + 3x - 5

Tips: powers need ^ — type x^2 for x² (just x2 means x·2). Multiplication is implicit — 2x, 3sin(x) and 5(x+1) all work. Use sqrt(x) for √x, pi for π and e for Euler's number. The expression is treated as ∫ f(x) dx.

x² 3x² + 2x − 5 (2x+1)³ sin(x) cos(2x+1) eˣ e^(3x) 1/x ln(x) √x

Definite integral — optional

Lower bound a

Upper bound b

Leave both blank for the indefinite integral. Fill both to compute F(b) − F(a) and the numeric area under the curve.

Antiderivative

About the Calculus Integral Calculator

The Calculus Integral Calculator finds the antiderivative of a function with respect to x and shows the working line by line. It is built for students, educators and self-learners who want to see why an answer is correct, not just the final result. Every step names the rule that was applied — power rule, sum rule, constant-multiple rule, u-substitution with a linear inner function, trigonometric rules, exponential rule or integration by parts — so you can match it back to your textbook or lecture notes.

All calculations run entirely in your browser. No data is sent to a server, no sign-up is required, and the tool is GDPR compliant.

What is integration?

Integration is the reverse of differentiation. If F'(x) = f(x), then F(x) is called an antiderivative of f(x), written:

∫ f(x) dx = F(x) + C

The constant C represents the family of curves that all share the same derivative. For a definite integral with limits a and b, you evaluate the antiderivative at the bounds and subtract:

∫_a^b f(x) dx = F(b) − F(a)

This is the Fundamental Theorem of Calculus and gives the signed area under the curve y = f(x) between x = a and x = b.

Standard integration rules

Constant rule: ∫ k dx = k·x + C
Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1
Reciprocal rule: ∫ 1/x dx = ln|x| + C
Exponential rule: ∫ e^x dx = e^x + C and ∫ a^x dx = a^x/ln(a) + C
Sum / difference rule: ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
Constant-multiple rule: ∫ k·f(x) dx = k · ∫ f(x) dx
Trig rules: ∫ sin(x) dx = −cos(x), ∫ cos(x) dx = sin(x), ∫ sec²(x) dx = tan(x)
u-substitution (linear inner): ∫ f(ax + b) dx = (1/a) · F(ax + b)
Integration by parts: ∫ ln(x) dx = x·ln(x) − x + C

How to use this calculator

Type or click your integrand using the keyboard above. Use x as the variable, ^ for powers and standard operators.
Press Integrate. The big result panel shows the antiderivative with + C, plus a numbered breakdown of every rule applied.
(Optional) Open the Definite Integral panel and type the lower and upper bounds. The calculator evaluates F(b) − F(a) and shows the numeric area.
Try the example chips to see how common integrals are solved — polynomials, trig, exponentials, logarithms and reciprocals.

Common uses

GCSE Further Maths, A-Level Maths and IB HL homework
Leaving Certificate higher-level calculus problems
First-year university calculus courses (engineering, physics, economics, computer science)
Quick antiderivative checks while studying or marking
Building intuition for the Fundamental Theorem of Calculus

Frequently Asked Questions

Why does my answer have a "+ C" at the end? Indefinite integrals are determined only up to an additive constant — any constant disappears under differentiation. The "+ C" is a reminder that infinitely many functions share the same derivative.

Can it integrate with respect to a variable other than x? The calculator uses x as the variable. If your problem uses a different letter, mentally substitute it for x before entering. The structure of the answer is the same.

How accurate is the numeric definite integral? When a closed-form antiderivative is found, F(b) − F(a) is evaluated with double-precision floating-point arithmetic, accurate to about 12–13 decimal places. When no closed form is available, the calculator falls back to Simpson's rule with adaptive subdivision, which is accurate to roughly 1×10⁻⁸ for smooth integrands.

Why does some integrand return "not supported"? The tool focuses on the standard rules taught in introductory calculus: power, sum/difference, constant-multiple, trigonometric, exponential, logarithmic and u-substitution with a linear inner function. Integrals that need integration by parts on products like x·sin(x), trigonometric substitution, partial fractions on non-linear denominators, or special functions are intentionally outside its scope. Try splitting the integrand into simpler terms first, or use the related Graphing Calculator to visualise the area numerically.

Does it handle constants like π and e? Yes. Type pi or click the π key for π, and e for Euler's number. They are treated as exact constants in the antiderivative and converted to numbers when evaluating definite bounds.

Can I use this for negative or fractional powers? Yes. The power rule works for any real n ≠ −1 — try x^(-2), x^(1/2) or sqrt(x).

Is my data stored anywhere? No. Every calculation runs locally in your browser and nothing is uploaded.

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Important Note: This tool is intended to provide estimates and step-by-step educational explanations and should not be used as a substitute for professional advice. Information generated by these calculators may be incomplete and does not account for all individual circumstances. Always seek the counsel of a certified expert (such as a financial advisor, healthcare provider, or licensed engineer) before taking action based on these results.

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