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Integral calculator

Calculate an integral

Enter an integrand, variable, and optional bounds, then follow the method from recognition to a checked result.

Use the math keyboard or your device keyboard.
Try an example

Result

Antiderivative

12ex2+C\frac12e^{x^2}+C
Conditions
  • x is real.
  • C is an arbitrary constant.

Steps

  1. Recognize the inner derivative. The derivative of x squared is proportional to the remaining factor x.u=x2u=x^2
  2. Rewrite the differential. Replace x dx with one half du.x dx=12dux\,dx=\frac12du
  3. Integrate and substitute back. Integrate e to the u and restore x squared.12ex2+C\frac12e^{x^2}+C
Independent check

Differentiating the antiderivative with the chain rule returns x times e to the x squared.

Scope

What this integral calculator covers

The integral calculator distinguishes an antiderivative family from a definite accumulated value. It keeps the constant of integration, bounds, and method visible so the result can be checked.

Basic antiderivatives

Integrate sums of supported powers, exponentials, and trigonometric terms term by term.

Examples: polynomial sums, e to the x, sine and cosine

Substitution patterns

Recognize a composite expression accompanied by a constant multiple of its derivative.

Examples: x times e to the x squared, x over x squared plus one

Integration by parts

Exchange a product integral for a simpler one when differentiating one factor makes progress.

Examples: x times e to the x, x times ln x

Definite integrals

Evaluate a verified antiderivative at both endpoints and subtract in upper-minus-lower order.

Examples: signed area, net accumulation

How to use it

Enter enough information for one clear task

  1. 1
    Enter the integrand

    Use parentheses around exponents, products, and denominators so the mathematical structure is clear.

  2. 2
    Choose indefinite or definite

    For a definite integral, supply both bounds and keep the variable consistent.

  3. 3
    Review the selected method

    Check whether substitution, integration by parts, or a direct rule actually makes the remaining integral simpler.

  4. 4
    Verify the result

    Differentiate an antiderivative or compare a definite value with independent numerical quadrature.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Power rule

power antiderivative

∫x4 dx\int x^4\,dx

Expected result

x55+C\frac{x^5}{5}+C

Exponential

reverse chain rule

∫e3x dx\int e^{3x}\,dx

Expected result

13e3x+C\frac13e^{3x}+C

Substitution

u-substitution

∫xex2 dx\int xe^{x^2}\,dx

Expected result

12ex2+C\frac12e^{x^2}+C

Substitution

logarithmic pattern

∫2xx2+1 dx\int\frac{2x}{x^2+1}\,dx

Expected result

ln⁡(x2+1)+C\ln(x^2+1)+C

Integration by parts

parts

∫xex dx\int xe^x\,dx

Expected result

ex(x−1)+Ce^x(x-1)+C

Definite integral

endpoint evaluation

∫0πsin⁡x dx\int_0^{\pi}\sin x\,dx

Expected result

22

Complete example

Integrate x times ln x by parts

A logarithm becomes simpler when differentiated, while x is straightforward to integrate. That makes the split useful on x greater than zero.

∫xln⁡(x) dx\int x\ln(x)\,dx
  1. 1
    Choose u and dv

    Differentiate the logarithm and integrate x.

    u=ln⁡x,dv=x dxu=\ln x,\quad dv=x\,dx
  2. 2
    Find du and v

    Compute the paired differential and antiderivative.

    du=1x dx,v=x22du=\frac1x\,dx,\quad v=\frac{x^2}{2}
  3. 3
    Apply the formula

    Substitute into integral u dv equals uv minus integral v du.

    x22ln⁡x−12∫x dx\frac{x^2}{2}\ln x-\frac12\int x\,dx
  4. 4
    Finish the simpler integral

    Integrate x and retain the arbitrary constant.

    x22ln⁡x−x24+C\frac{x^2}{2}\ln x-\frac{x^2}{4}+C
∫xln⁡(x) dx=x22ln⁡(x)−x24+C\int x\ln(x)\,dx=\frac{x^2}{2}\ln(x)-\frac{x^2}{4}+C

Verification: Differentiating the two terms gives x ln x plus x over 2 minus x over 2, leaving the original integrand.

Avoidable errors

Common mistakes and how to fix them

Missing the constant

Problem: Stopping an indefinite integral without plus C.

Why it matters: Every constant has derivative zero, so antiderivatives form a family.

Better approach: Add C after the final indefinite antiderivative.

Using the power rule at n = -1

Problem: Dividing by n plus 1 when that quantity is zero.

Why it matters: The power formula excludes x to the negative one.

Better approach: Use the logarithmic antiderivative: integral 1 over x dx equals ln absolute x plus C.

Reversing endpoints

Problem: Computing lower minus upper.

Why it matters: The Fundamental Theorem requires F(b) minus F(a).

Better approach: Write the endpoint substitution before simplifying signs.

Choosing a harder substitution

Problem: Selecting u without checking whether the remaining differential is present.

Why it matters: The transformed integral may be no simpler or may still contain x.

Better approach: Differentiate the proposed u and confirm the remaining factors can be replaced.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • Simplify the result independently and compare equivalent forms.
  • Substitute regular sample values when a numerical check is meaningful.
  • Check the domain and any one-sided, endpoint, or constant assumptions.

When to stop and revise the input

  • Revise an expression when the variable, boundary, direction, or grouping is ambiguous.
  • Treat a symbolic result as unsupported when the engine cannot establish an elementary form.
  • Use the steps to check work, not as a substitute for course or examination rules.

Common questions

Calculate an integral FAQ

Does the calculator show steps?

Yes. A result includes the interpreted problem, the final answer, the mathematical operation used, relevant conditions, and a verification note when an independent check is available.

Why can a correct answer look different from a textbook answer?

Equivalent algebraic and trigonometric forms can look different while representing the same function. Indefinite integrals can also differ by a constant. Compare by simplifying, differentiating, or substituting values within the shared domain.

What should I include in the input?

State the expression, variable, and operation. Add an interval, upper and lower bounds, a one-sided direction, derivative order, or initial condition whenever the task depends on it.

How should I check an important result?

Inspect the recognized problem and assumptions first. Then use an independent check such as differentiating an antiderivative, substituting a solution into the original equation, or comparing one-sided numerical values.

Continue learning

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.