Integral Calculator

Calculate integrals with algebraic parameters, roots, powers, exponential, logarithmic, trigonometric and other supported functions. Solution coverage depends on the symbolic engine.

Type the function only (do not type ∫ or dx). Use parentheses to show the full denominator or root: 1/(1+x^2) + 12/sqrt(1-x^2).

Math preview
\(f(x)=x^2\)
Mathematical symbols & function keypad

Tap a key to insert it at the cursor. Fraction and root inputs must be grouped with parentheses: type 1/(1+x^2) and 12/sqrt(1-x^2), not 1/1+x^2 or 12/√1-x^2.

Algebra symbols, numbers, powers, roots and operators.

Numbers & basic algebra
Algebraic forms
Operators
Exponential & logarithms
Greek parameters (engine support varies)
What do these symbols mean?

+ Add terms. − Subtract terms. * Multiply. / Divide.

^ Power: x^3 means x cubed. sqrt(x) is the square root; root(n,x) is the nth root.

sin, cos, tan are trigonometric functions (angles in radians). sec, csc, cot are their reciprocal functions.

asin, acos, atan are inverse trigonometric functions. They are not the same as 1/sin, 1/cos, or 1/tan.

sinh, cosh, tanh are hyperbolic functions. exp(x) means e to the power x.

ln(x) is the natural logarithm. log(x) is the logarithm supported by the engine. abs(x) is absolute value.

pi is π (about 3.14159); e is Euler's number (about 2.71828); i is the imaginary unit when complex support is available. erf, Ei, Si, Ci, Γ and W denote special functions; they may not be supported by this browser engine.

∞ denotes infinity in mathematics, but this build's bounds currently require finite numeric values. dx denotes the differential; enter only the integrand, not dx. Greek letters can be used as parameters only where the parser supports them.

∫ means integration; + C is the arbitrary constant for an indefinite integral. Definite integrals require lower and upper limits.

Try an example

Function Graph

The graph shows the integrand. For definite integrals, the interval is highlighted when supported.

3,000 Practice Questions — 6 Categories

Select an integration category, then choose a practice question. The expression and recoverable limits will load automatically.

Download all 3,000 practice questions with solutions (PDF)

Integral Calculator Guide

How Free Integral Calculator Works?

An integral calculator is an online mathematics tool that helps students, teachers, engineers, and other users solve integration problems. It can calculate supported indefinite and definite integrals, making calculus easier to practise and understand. Instead of solving every expression manually, you can enter a mathematical function, choose the integration variable, and click the Calculate Integral button. The calculator processes your expression and displays the result. Depending on the expression and the calculator's supported rules, it may also provide solution steps and a graph. In this guide, you will learn how an integral calculator works, how to use its features, and how to enter numbers and mathematical expressions correctly.

What Is an Integral Calculator?

An integral calculator is a digital tool designed to solve integration problems. Integration is a fundamental operation in calculus. It is commonly used to find antiderivatives, calculate accumulated quantities, and determine the signed area under a curve.

For example, consider the following integral:

\int x^2,dx

Its answer is:

\frac{x^3}{3}+C

Here, (x) is the integration variable, and (C) represents the constant of integration. An online integral calculator performs the supported mathematical operations and presents the answer without requiring you to calculate every step manually.

How Does an Integral Calculator Work?

An integral calculator generally works through three main stages.

__TRIG_HTML_BLOCK__ Steps to Calculate a Trigonometric Integral

Enter the trigonometric expression, such as cos(3*x), in the input field. Select x as the integration variable. Choose an indefinite integral if you do not have limits. Click the Calculate button to view the result. Review the solution steps if they are available for your selected expression.

Tip: Use explicit multiplication symbols, such as 2*x*cos(x^2), when entering expressions. Calculator support depends on the functions and integration rules implemented in the tool.

1. Enter a Mathematical Expression

First, enter the function you want to integrate. You can use numbers, variables, powers, and supported mathematical functions.

For example:

x^2

This expression means (x) raised to the power of two.

2. Select the Integration Variable

Choose the variable with respect to which you want to integrate. Common variables include (x), (t), and (y).

For example, in an integral written as (\int x^2,dx), the variable is (x). The notation (dx) indicates integration with respect to (x).

3. Calculate and Review the Result

Click the calculate integrals button. The calculator sends the expression to its mathematical engine, which attempts to find an antiderivative or evaluate a definite integral. The result is then displayed using mathematical notation.

The available result depends on the expression and the calculation engine's capabilities. Some complicated integrals may require alternative methods or a more advanced mathematical solver.

How to Use Our integration calculator ?

Follow these instructions to use the calculator on this page.

Enter the function: Type your expression into the Function field. For example, enter x^2.

Choose the variable: Select x, t, or y from the Integration variable menu.

Select the integral type: Choose Indefinite integral or Definite integral.

Enter the limits: For a definite integral, provide the lower and upper bounds.

Calculate the integral: Click the Calculate Integral button.

Review the answer: Check the result and any available solution steps or graph.

For the first test, enter x^2, select x, and choose Indefinite integral. The expected answer is (x^3/3+C).

How to Enter Numeric Digits in an Integral Calculator ?

Numbers are an important part of mathematical expressions. You can combine numeric digits with variables and mathematical operators to create different functions.

1. Enter a Whole Number

A whole number can be used by itself or multiplied by a variable.

Examples:

5

3*x
7*x^2

The expression 3*x means three multiplied by (x). The expression 7*x^2 means seven multiplied by (x) squared.

2. Enter Decimal Numbers

Use a decimal point to enter a decimal number.

Examples:

2.5*x
0.75*x^2
3.14*x

For most calculators, the decimal point is written as a period, not a comma. Enter 2.5*x rather than 2,5*x.

3. Enter Negative Numbers

Use the minus sign to enter a negative number.

Examples:

-x

-3*x^2
x^2-4*x

A negative coefficient changes the sign of the corresponding term. For example, the integral of (-3x^2) is (-x^3+C).

4. Enter Powers and Exponents

Use the caret symbol ^ to indicate a power.

Examples:

Power notation examples

Use x^2 for x squared, x^3 for x cubed, 2*x^4 for two times x to the fourth power, and x^0.5 for the square-root power where defined.

For better compatibility, use x^2 instead of typing a superscript character such as x².

5. Enter Fractions

Use the forward slash / to divide one expression by another.

Examples:

1/x
3*x/(x^2+1)
(x+1)/(x-1)

Use parentheses to group the numerator and denominator correctly. For example, 3*x/(x^2+1) clearly tells the calculator to divide the entire expression (3x) by (x^2+1).

6. Enter Multiplication Correctly

For the best compatibility with this calculator, use the asterisk * for multiplication.

For example, write:

3*a*x/(c^2*x^2+b^2)

This represents:

\frac{3ax}{c^2x^2+b^2}

Do not include dx in the Function field. Select the integration variable separately using the calculator's variable menu.

How to Use Indefinite Integrals

An indefinite integral finds a family of antiderivatives without specified lower or upper limits. Its integral solver includes the constant of integration (C).

For example:

\int 4x^3,dx=x^4+C

To calculate this integral, enter 4*x^3, select x, and choose Indefinite integral.

The calculator should return the antiderivative if the expression is supported. Remember that the constant (C) is important because adding any constant does not change the derivative.

How to Use Definite Integrals and Numeric Limits ?

A definite integral has a lower limit and an upper limit. These limits define the interval over which the function is integrated.

For example:

\int_0^2 x^2,dx=\frac{8}{3}

To enter this problem:

Type x^2 in the Function field.

Select x as the integration variable.

Choose Definite integral.

Enter 0 as the lower bound.

Enter 2 as the upper bound.

Click Calculate Integral.

The correct numerical answer is approximately (2.666667).

The limits are numeric values, while the integration variable represents the variable used in the function. If the limits are reversed, the definite integral changes sign.

Common Mistakes to Avoid

When using an antiderivative calculator, small input errors can prevent the expression from being interpreted correctly.

Using an unclear multiplication format: Prefer 3*x over 3x.

Entering the differential in the function: Type x^2, not x^2 dx.

Missing parentheses: Use 3*x/(x^2+1) when the entire denominator must be included.

Using the wrong integration variable: Ensure the selected variable matches the variable in the expression.

Confusing powers with multiplication: x^2 means (x) squared, while 2*x means twice (x).

Entering incorrect limits: Check the lower and upper bounds before calculating a definite integral.

Always verify complex results, especially when using an online tool for academic or technical work.

Frequently Asked Questions

Is an integral calculator easy to use?

Yes. Enter a supported function, select the integration variable and integral type, add limits if required, and click Calculate Integral.

How do I enter a power such as x squared?

Use the caret symbol. Enter x^2 to represent (x) squared.

How do I enter a decimal coefficient?

Use a decimal point and the multiplication symbol. For example, enter 2.5*x.

What is the difference between definite and indefinite integration?

An indefinite integral produces an antiderivative and includes a constant of integration. A definite integral uses specified limits and returns a numerical value when the integral is defined and can be evaluated.

Why might my integral not calculate?

The expression may contain unsupported syntax, the mathematical engine may not support that particular integral, or a required script may not have loaded. Try a simpler expression and use explicit multiplication signs and parentheses.

Does the calculator always show step-by-step solutions?

Not necessarily. Detailed steps are available only for integration rules that the current version recognizes. Some expressions may return an answer without a complete derivation.

Conclusion

An integral calculator with steps helps make integration more convenient by processing supported mathematical expressions and displaying their results. By learning how to enter numeric digits, decimal coefficients, powers, fractions, and integration limits, you can reduce input errors and practise calculus more effectively. Start with simple functions such as x^2, then gradually try more advanced expressions as you become familiar with the calculator.