# How to write and evaluate indefinite integrals

## Description

The antiderivative of a function is expressed using an indefinite integral, as in

\[\int f(x)\;dx.\]How can we write and evaluate indefinite integrals using software?

Related tasks:

- How to compute the derivative of a function
- How to write and evaluate definite integrals
- How to write and evaluate Riemann sums

## Using SymPy, in Python

This answer assumes you have imported SymPy as follows.

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from sympy import * # load all math functions
init_printing( use_latex='mathjax' ) # use pretty math output

Let’s choose an example formula whose antiderivative we will compute.

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var( 'x' )
formula = 3*sqrt(x)
formula

$\displaystyle 3 \sqrt{x}$

Use the `Integral`

function to build a definite integral without evaluating it.
The second parameter is the variable with respect to which you’re integrating.

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Integral( formula, x )

$\displaystyle \int 3 \sqrt{x}\, dx$

Use the `integrate`

function to perform the integration, showing the answer.

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integrate( formula, x )

$\displaystyle 2 x^{\frac{3}{2}}$

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integrate( formula, x ) + var('C') # same, but with a constant of integration

$\displaystyle C + 2 x^{\frac{3}{2}}$

Content last modified on 24 July 2023.

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