Integral Calculator
With this calculator, you can evaluate indefinite or definite integrals. Examples of calculations can be found in the corresponding section.
Some basic integrals can be found without calculation by looking up the antiderivative in the table.
Antiderivative
An antiderivative for a function $f(x)$ is a function $F(x)$ whose derivative is equal to $f(x)$, that is, $F^{\prime}(x) = f(x)$. Finding an antiderivative is the operation inverse to differentiation.
If $F(x)$ is an antiderivative for $f(x)$, then the function $F(x) + C$, where $C$ is an arbitrary constant, is also an antiderivative for $f(x)$.
Indefinite Integral
The indefinite integral for a function $f(x)$ is the set of all antiderivatives of this function. It is denoted as follows:
$$\int f(x) dx = F(x) + C,$$
where
- $\int$ is the integral sign
- $f(x)$ is the integrand
- $dx$ is the differential element
- $F(x)$ is the antiderivative
- $C$ is the constant of integration
The operation of finding the integral is called integration.
Properties
The main properties of the indefinite integral:
$$\int a \cdot f(x) dx = \\ = a \cdot \int f(x) dx \ \ \ (a \ne 0)$$
$$\int (f(x) \pm g(x)) dx = \\ = \int f(x) dx \pm \int g(x) dx$$
Calculation Examples
Below are examples of calculating indefinite integrals. To perform these calculations on the integral calculator, you need to sequentially click on the buttons indicated under each example. Note: enter int into the empty field below the calculator screen using your computer’s keyboard.
$$\int x^3 dx = \frac {x^4} {4} + C$$
i n t ( x xy 3 ) =
$$\int \sin 7x dx = - \frac {\cos 7x} {7} + C$$
i n t ( sin 7 x ) =
$$\int x^3 y dy = \frac {x^3 y^2} {2} + C$$
i n t ( x x^3 y , y ) =
Integral Table
Table of basic indefinite integrals and their corresponding antiderivatives:
| $\int f(x) dx$ | $F(x) + C$ |
|---|---|
| $$\int 0 \cdot dx$$ | $$C$$ |
| $$\int dx$$ | $$x + C$$ |
| $$\int x^n dx$$ | $$\frac {x^{n + 1}} {n + 1} + C \ \ \ (n \ne -1)$$ |
| $$\int \frac {1} {x} dx$$ | $$\ln | x | + C$$ |
| $$\int e^x dx$$ | $$e^x + C$$ |
| $$\int a^x dx$$ | $$\frac {a^x} {\ln a} + C \ \ \ (a > 0, a \ne 1)$$ |
| $$\int \cos x dx$$ | $$\sin x + C$$ |
| $$\int \sin x dx$$ | $$- \cos x + C$$ |
| $$\int \frac {dx} {\cos^2 x}$$ | $$\tg x + C$$ |
| $$\int \frac {dx} {\sin^2 x}$$ | $$- \ctg x + C$$ |
| $$\int \frac {dx} {\sqrt {1 - x^2}}$$ | $$\begin{gathered} \arcsin x + C_1 = \\ = - \arccos x + C_2 \\ (C_2 = \frac {\pi} {2} + C_1) \end{gathered}$$ |
| $$\int \frac {dx} {1 + x^2}$$ | $$\arctg x + C$$ |
| $$\int \ch x dx$$ | $$\sh x + C$$ |
| $$\int \sh x dx$$ | $$\ch x + C$$ |
Definite Integral
If $F(x)$ is an antiderivative for the function $f(x)$, which is defined and continuous on the interval $[a;b]$, then the definite integral is calculated by the formula:
$$\int _a^b f(x) dx = F(x) \mid _a^b = F(b) - F(a)$$
Properties
The main properties of the definite integral:
$$\int _a^a f(x) dx = 0$$
$$\int _a^b f(x) dx = - \int _b^a f(x) dx$$
$$\int _a^b f(x) dx = \int _a^c f(x) dx + \\ + \int _c^b f(x) dx \ \ \ (c \in [a;b])$$
Calculation Examples
Below are examples of calculating definite integrals. To perform these calculations on the calculator, you need to sequentially click on the buttons indicated under each example. Note: enter int into the empty field below the calculator screen using your computer’s keyboard.
$$\int _1^3 (5 + x) dx = 14$$
i n t ( 5 + x ,
x 2nd var 1 . . 3 2nd ) =
$$\int _5^8 x^2 dx = 129$$
i n t ( x x2 ,
x 2nd var 5 . . 8 2nd ) =