Integral Calculator

Find an indefinite integral, follow the chosen method, and keep the constant of integration.

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Enter an indefinite integral, such as integrate x e^(x^2) dx

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An integral is a family of antiderivatives

An indefinite integral asks for every function whose derivative equals the integrand. Those functions differ by a constant, which is why a complete answer ends with + C.

The right method depends on the integrand. A disguised chain rule suggests substitution, a product may call for integration by parts, and a rational expression may need algebra or partial fractions before any integration rule is useful.

How to use the integral calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

The substitution pattern

When part of the integrand is an inner function and another part is its derivative, substitution can turn the problem into a standard form.

\int f (g (x)) g^{'} (x) d x = F (g (x)) + C

A reliable way to work through it

Match a standard form

Check first for the power, logarithm, exponential, and trigonometric antiderivative rules.

Choose a transformation

Use substitution, integration by parts, identities, or partial fractions only when the structure supports it.

Differentiate the result

A quick derivative should simplify back to the original integrand on the stated domain.

Worked example

Evaluate ∫x e^(x²) dx

u = x^{2}, d u = 2 x d x

Choose the exponent as the substitution.

x d x = \frac{1}{2} d u

Solve for the remaining differential factor.

\int x e^{x^{2}} d x = \frac{1}{2} \int e^{u} d u

Rewrite the integral in terms of u.

\frac{1}{2} e^{x^{2}} + C

Integrate and substitute x² back for u.

The antiderivative is ½e^(x²) + C. Differentiating it returns xe^(x²).

Common mistakes to check

Leaving off the constant

An indefinite integral represents a family of antiderivatives, so the final answer needs + C.

Using the power rule on 1/x

The exponent -1 is the exception. Its real antiderivative is ln|x| + C.

Forcing an elementary answer

Some integrals require special functions or have no elementary antiderivative.

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Questions students ask

Why does an indefinite integral need + C?

Differentiation removes constants, so every antiderivative belongs to a family whose members differ by a constant.

How can I check an antiderivative?

Differentiate it. The result should simplify to the original integrand wherever both expressions are defined.

How is this different from a definite integral?

An indefinite integral returns a family of functions. A definite integral uses bounds and returns a value.

What if no elementary antiderivative exists?

A correct result may use a named special function or leave the integral unevaluated instead of inventing an elementary form.