How to Perform Indefinite (Symbolic) Integration in Python Using SymPy ?

Introduction

You can perform indefinite integration (symbolic integration) easily in Python using SymPy, which is a symbolic mathematics library.

Step 1. Import the necessary modules

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import sympy as sp

Step 2. Define the symbol(s)

You first define the variable of integration using sp.Symbol or sp.symbols:

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x = sp.Symbol('x')

Step 3. Define the function to integrate

Let’s say we want to integrate ( f(x) = x^2 * e^x ):

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f = x**2 * sp.exp(x)

Step 4. Compute the indefinite integral

Use sp.integrate():

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F = sp.integrate(f, x)
print(F)

Output:

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x**2*exp(x) - 2*x*exp(x) + 2*exp(x)

(SymPy omits the “+ C” constant by default.)

Step 5. (Optional) Add the constant of integration

If you want to display it explicitly:

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C = sp.Symbol('C')
F_with_C = F + C
print(F_with_C)

Example with multiple variables

If your expression has multiple variables, you can specify which one to integrate with respect to:

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y = sp.Symbol('y')
expr = x**2 + sp.sin(y)
integral_x = sp.integrate(expr, x)
integral_y = sp.integrate(expr, y)

print("∫ wrt x:", integral_x)
print("∫ wrt y:", integral_y)

Example: Trigonometric integral

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f = sp.sin(x) * sp.cos(x)
F = sp.integrate(f, x)
print(F.simplify())

Output:

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sin(x)**2/2

Rendering SymPy Expressions in LaTeX (Jupyter Notebook)

In a Jupyter notebook, you can render SymPy expressions beautifully in LaTeX using the built-in display tools.
This allows symbolic results such as integrals or equations to appear as properly formatted mathematical expressions, improving readability and presentation quality.

Option 1: Using sp.init_printing() (recommended)

This automatically makes all SymPy outputs render in LaTeX format.

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import sympy as sp

sp.init_printing()   # enables nice LaTeX output

x = sp.Symbol('x')
f = x**2 * sp.exp(x)

F = sp.integrate(f, x)
F

Output:

You’ll see something like

nicely rendered as LaTeX in the notebook output cell.

Option 2: Use display() and sp.Eq()

If you want to show an equation-style output:

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from sympy import symbols, integrate, exp, Eq
from IPython.display import display

x = symbols('x')
f = x**2 * exp(x)
F = integrate(f, x)

display(Eq(sp.Integral(f, x), F))

Output:

Option 3: Use display(Math(...)) explicitly

For full manual control of LaTeX rendering:

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from IPython.display import display, Math

display(Math(f"\\int x^2 e^x \\,dx = {sp.latex(F)} + C"))

This uses SymPy’s latex() function to convert symbolic expressions into valid LaTeX strings.

Option 4: (Optional) Inline LaTeX with Markdown cell

You can also insert a Markdown cell and use something like:

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$\int x^2 e^x \,dx = e^x(x^2 - 2x + 2) + C$

for presentation-quality output.

References