# How to write and evaluate indefinite integrals (in Python, using SymPy)

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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?

## Solution

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


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