# antiderivative calculator

## Calculus antiderivative

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### Function : antiderivative

#### Summary :

The antiderivative calculator allows to calculate a primitive online with detail and calculation steps.

Antiderivative online

#### Description :

The antiderivative calculator allows to find primitive function, primitive integral or indefinite integral of a function using integration properties and different calculation mechanisms online. The antiderivative calculator is able to do symbolic antidifferentiation.

# Calculate antiderivatives

## Calculate online an antiderivative of a polynomial

The antiderivative calculator allows to integrate online any polynomial.

For example, to compute an antiderivative of the polynomial following x^3+3x+1, you must enter antiderivative(x^3+3x+1;x), after calculating the result (3*x^2)/2+(x^4)/4+x is returned.

## Calculate online usual functions antiderivatives

The antiderivative calculator is able to calculate online all antiderivatives of usual functions : sin, cos, tan, ln, exp, sh, th, sqrt (square root), and many more ...

So, to obtain an antiderivative of the cosine function with respect to the variable x, type, antiderivative(cos(x);x), result sin(x) is returned after calculation..

## Calculate online a function sum

Integration is a linear function, using this property allows the function to get the required result.

For the online calculation of antiderivative of function sum, simply type the mathematical expression that contains the sum, specify the variable and apply function .

For example, to calculate an antiderivative line of the sum of the following functions cos(x)+sin(x)type antiderivative(cos(x)+sin(x);x), after calculating the result sin(x)-cos(x) is returned.

## Online integral of a function difference

To calculate online an antiderivative of a difference of function, just input mathematical expression that contains the difference, specify variable and apply function antiderivative.

For example, to calculate online an antiderivative of the difference of the following functions cos(x)-2x type antiderivative(cos(x)-2x;x), after calculating the result sin(x)-x^2 is displayed.

## Function composition online integral

To calculate online an antiderivative of composition of functions of the form u(ax+b), where u is a usual function, simply type mathematical expression that contains the function, specify variable and apply function antiderivative.

For example, to calculate online an antiderivative of the following function exp(2x+1) you must enter antiderivative(exp(2x+1);x), after calculating the result exp(2x+1)/2 is displayed.

For example, to calculate online an antiderivative of the following function sin(2x+1) you must enter antiderivative(sin(2x+1);x), to get the following result -cos(2*x+1)/2.

## Integration by parts

For calculation of some functions, calculator is able to use integration by parts. The formula used is as follows : Let f and g be two continuous functions, int(f'g)=fg-int(fg')

For example, to calculate an antiderivative x*sin(x), calculator uses the integration by parts, to get the result, you must enter antiderivative(x*sin(x);x), after calculation, result sin(x)-x*cos(x) is returned with steps and detailed calculations.

# How to integrate a function?

To integrate a function, the following formulas can be used and the usual calculation rules applied:

Table of primitives functions
f(x)F(x)
kkx+c
xx^2/2+c
x^nx^(n+1)/(n+1)+c
sqrt(x)-1/((n-1)*x^(n-1))+c
abs(x)x/2+c
arccos(x)x*arccos(x)-sqrt(1-(x)^2)+c
arcsin(x)x*arcsin(x)+sqrt(1-(x)^2)+c
arctan(x)x*arctan(x)-1/2*ln(1+(x)^2)+c
ch(x)sh(x)+c
cos(x)sin(x)+c
cotan(x)ln(sin(x))+c
coth(x)ln(sh(x))+c
exp(x)exp(x)+c
ln(x)x*ln(x)-x+c
log(x)(x*log(x)-x)/ln(10)+c
racine_cubique(x)3/4*(x)^(4/3)+c
sh(x)ch(x)+c
sin(x)-cos(x)+c
sqrt(x)2/3*(x)^(3/2)+c
tan(x)-ln(cos(x))+c
th(x)ln(ch(x))+c
The following conventions are used in the primitive integral table: c represents a constant; F'(x)=f(x).

By applying the integration formulas and using the table of usual antiderivatives, it is possible to calculate many function primitives integral. These are the calculation methods used by the calculator to find the indefinite integral.

The antiderivative calculator allows to calculate a primitive online with detail and calculation steps.

#### Syntax :

antiderivative(fonction;variable), function is the function to integrate.

#### Examples :

##### Antidifferentiation of a trigonometric function

This example shows how to use the antiderivative calculator to integrate sin (x) + x with respect to x, you must enter:

Calculate online with antiderivative (antiderivative calculator )