Integral Calculator

Work out definite and indefinite integrals, with the technique named and every step shown.

Try an example:

What an integral is

Integration reverses differentiation. An indefinite integral $\int f (x) d x$ is the family of functions whose derivative is $f (x)$ , written with a constant $+ C$ . A definite integral $\int_{a}^{b} f (x) d x$ is a number: the signed area between the curve and the $x$ -axis from $a$ to $b$ .

How to use the integral calculator

Type or paste your problem into the box above — for example ∫ (3x^2 + 2x) dx. Use ^ for powers, / for fractions and sqrt() for square roots, or describe it in words.
Press Enter. The solver works through the problem with every step shown; switch Steps off if you only want the final answer.
Ask a follow-up in the same chat — "why did you do that?" or "now check it" — and it answers with the earlier working in context.

The fundamental theorem of calculus

If $F$ is any antiderivative of $f$ , then

\int_{a}^{b} f (x) d x = F (b) - F (a)

So a definite integral is usually found by working out an antiderivative first, then evaluating it at the two limits.

Choosing a technique

Basic rules: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ for $n \neq = - 1$ , and $\int \frac{1}{x} d x = ln ∣ x ∣ + C$ .
Substitution: when the integrand contains a function and its derivative, like $2 x cos (x^{2})$ .
Integration by parts: for products such as $x e^{x}$ , using $\int u d v = uv - \int v d u$ .
Partial fractions: for rational functions whose denominator factors.

Worked example: $\int x e^{x} d x$

Choose $u$ and $d v$ Let $u = x$ , because it gets simpler when differentiated, and $d v = e^{x} d x$ .
Find $d u$ and $v$ $d u = d x$ and $v = e^{x}$ .
Apply integration by parts $\int x e^{x} d x = x e^{x} - \int e^{x} d x$
Finish and add the constant $x e^{x} - e^{x} + C = e^{x} (x - 1) + C$
Check by differentiating $\frac{d}{d x} [e^{x} (x - 1)] = e^{x} (x - 1) + e^{x} = x e^{x}$

Common mistakes to avoid

Dropping the $+ C$

Every indefinite integral needs the constant of integration, because there are infinitely many antiderivatives.

Using the power rule on $\frac{1}{x}$

$\int x^{- 1} d x$ is $ln ∣ x ∣ + C$ . The power rule would divide by zero.

Forgetting to change the limits

With substitution in a definite integral, either convert the limits to the new variable or switch back to $x$ before evaluating.

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Questions students ask

Can it solve definite integrals?

Yes. Give it the limits — integrate x^3 from 0 to 2 — and it finds the antiderivative and evaluates it: $[\frac{x ^{4}}{4}]_{0}^{2} = 4$ .

Which techniques does it use?

Substitution, integration by parts, partial fractions, trigonometric identities and trigonometric substitution — whichever fits the integral. You can ask for a specific method.

Why is there a

+ C

in the answer?

Constants disappear when you differentiate, so any constant could have been there. $+ C$ stands for all of them.

Can it find the area between two curves?

Yes. Give it both functions and it finds where they intersect, then integrates the difference between them.