Comprehensive Integrals Calculator

Master integration with our powerful calculator suite. Compute indefinite, definite, multiple, and improper integrals with step-by-step solutions and visualizations.

Integrals Calculator

Our advanced integrals calculator handles a wide range of functions including polynomials, trigonometric, logarithmic, exponential, and more. Enter your function below to find its integral with detailed step-by-step solutions.

Indefinite Integrals
Definite Integrals
Multiple Integrals
Improper Integrals
Numerical Integration

Indefinite Integrals Calculator

Calculate the antiderivative (indefinite integral) of a function.

Use x as your variable. For example: x^2, sin(x), e^x, ln(x), etc.
The variable with respect to which the integral is calculated.

Indefinite Integral Result

Step-by-Step Solution

Examples

x^2 + 3x - 5
sin(x) + cos(x)
e^x
1/x
x * ln(x)

Definite Integrals Calculator

Calculate the definite integral of a function between specified limits.

Use x as your variable. For example: x^2, sin(x), e^x, ln(x), etc.
id="upper-limit" class="form-control" placeholder="e.g., 1">

Definite Integral Result

Step-by-Step Solution

Examples

∫₀¹ x² dx
∫₀ᵗ sin(x) dx
∫₀¹ e^x dx
∫₁² 1/x dx
∫₁ᵉ x ln(x) dx

Multiple Integrals Calculator

Calculate double and triple integrals of multivariable functions.

Use variables like x, y, z. For example: x^2 + y^2, sin(x)*cos(y), e^(x+y), etc.
Enter variables in the order of integration, separated by commas. For example: x,y for ∫∫ f(x,y) dy dx

Multiple Integral Result

Step-by-Step Solution

Examples

∫₀¹∫₀¹ xy dy dx
∫₀¹∫₀¹ (x² + y²) dy dx
∫₀ᵗ/²∫₀ᵗ/² sin(x)cos(y) dy dx
∫₀¹∫₀¹∫₀¹ xyz dz dy dx

Improper Integrals Calculator

Calculate integrals with infinite limits or discontinuities in the integrand.

Use x as your variable. For example: 1/x, e^(-x), 1/sqrt(x), etc.
Use "infinity" or "∞" for infinite limits. For example: 0 to infinity, -infinity to infinity, etc.

Improper Integral Result

Step-by-Step Solution

Examples

∫₀^∞ e^(-x) dx
∫₁^∞ 1/x² dx
∫₀¹ 1/√x dx
∫₋∞^∞ e^(-x²) dx
∫₀^∞ sin(x)/x dx

Numerical Integration Calculator

Approximate definite integrals using numerical methods.

Use x as your variable. Enter functions that may be difficult to integrate analytically.
More points generally lead to more accurate results, but require more computation.

Numerical Integration Result

Calculation Details

Examples

∫₀¹ sin(x²) dx
∫₀² √(1+x³) dx
∫₋₁¹ e^(-x²) dx
∫₀⁵ 1/(1+x²) dx
∫₁³ ln(x)/x dx

Common Integration Rules and Techniques

Understanding the basic rules and techniques of integration is essential for solving integrals. Here are the most important integration formulas with examples:

Basic Integration Rules

\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1 \\ \int dx = x + C \\ \int k \cdot f(x) dx = k \int f(x) dx

These are the fundamental rules for integrating powers of x and constant multiples.

Example: \int 3x^2 dx = 3 \cdot \frac{x^3}{3} + C = x^3 + C

Logarithmic Integration

\int \frac{1}{x} dx = \ln|x| + C

The integral of 1/x is the natural logarithm of the absolute value of x.

Example: \int \frac{2}{x} dx = 2\ln|x| + C

Exponential Integration

\int e^x dx = e^x + C \\ \int a^x dx = \frac{a^x}{\ln(a)} + C

The integral of an exponential function.

Example: \int e^{3x} dx = \frac{e^{3x}}{3} + C

Trigonometric Integration

\int \sin(x) dx = -\cos(x) + C \\ \int \cos(x) dx = \sin(x) + C \\ \int \tan(x) dx = -\ln|\cos(x)| + C

The integrals of the basic trigonometric functions.

Example: \int \sin(2x) dx = -\frac{\cos(2x)}{2} + C

Integration by Parts

\int u(x) v'(x) dx = u(x)v(x) - \int v(x) u'(x) dx

A technique used when integrating products of functions.

Example: \int x \cdot e^x dx = x \cdot e^x - \int e^x dx = x \cdot e^x - e^x + C

Integration by Substitution

\int f(g(x)) \cdot g'(x) dx = \int f(u) du, \quad \text{where } u = g(x)

A technique used to simplify complex integrals by substituting variables.

Example: \int \cos(x^2) \cdot 2x dx = \int \cos(u) du = \sin(u) + C = \sin(x^2) + C

Partial Fractions

\int \frac{P(x)}{Q(x)} dx = \int \left( \sum \frac{A_i}{(x-a_i)^{m_i}} + \sum \frac{B_ix + C_i}{(x^2 + bx + c)^{n_i}} \right) dx

A technique for integrating rational functions by decomposing them into simpler fractions.

Example: \int \frac{1}{x^2-1} dx = \int \left( \frac{1/2}{x-1} - \frac{1/2}{x+1} \right) dx = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right| + C

Definite Integral Properties

\int_a^b f(x) dx = -\int_b^a f(x) dx \\ \int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx \\ \int_a^b [f(x) \pm g(x)] dx = \int_a^b f(x) dx \pm \int_a^b g(x) dx

Important properties of definite integrals.

Example: \int_0^2 x dx + \int_2^4 x dx = \int_0^4 x dx = \frac{x^2}{2} \bigg|_0^4 = 8

Understanding Integrals and Their Applications

What is an Integral?

An integral represents the area under a curve, the accumulation of quantities, or the process of finding antiderivatives. There are two main types of integrals:

Types of Integrals

Applications of Integrals

Integrals have numerous applications across various fields:

Mathematics and Calculus

Physics

Engineering

Economics

Computer Science

Common Integration Formulas

Function f(x) Integral ∫f(x)dx
xn (n ≠ -1) xn+1/(n+1) + C
1/x ln|x| + C
ex ex + C
ax ax/ln(a) + C
sin(x) -cos(x) + C
cos(x) sin(x) + C
tan(x) -ln|cos(x)| + C
sec(x) ln|sec(x) + tan(x)| + C
1/(1+x2) arctan(x) + C
1/√(1-x2) arcsin(x) + C

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