In [ ]:
from sympy import *
init_printing(use_latex='mathjax')
n, m = symbols('n,m', integer=True)
x, y, z = symbols('x,y,z')


## Integrals¶

In the last section we learned symbolic differentiation with .diff. Here we'll cover symbolic integration with integrate.

Here is how we write the indefinite integral

$$\int x^2 dx = \frac{x^3}{3}$$
In [ ]:
# Indefinite integral
integrate(x**2, x)


And the definite integral

$$\int_0^3 x^2 dx = \left.\frac{x^3}{3} \right|_0^3 = \frac{3^3}{3} - \frac{0^3}{3} = 9$$
In [ ]:
# Definite integral
integrate(x**2, (x, 0, 3))


As always, because we're using symbolics, we could use a symbol whenever we previously used a number

$$\int_y^z x^n dx$$
In [ ]:
integrate(x**n, (x, y, z))


### Exercise¶

Compute the following integrals:

$$\int \sin(x) dx$$$$\int_0^{\pi} \sin(x) dx$$$$\int_0^y x^5 + 12x^3 - 2x + 1$$$$\int e^{\frac{(x - \mu)^2}{\sigma^2}}$$

Feel free to play with various parameters and settings and see how the results change.

In [ ]:
# Use integrate to solve the integrals above


Are there some integrals that SymPy can't do? Find some.

In [ ]:
# Use integrate on other equations.  Symbolic integration has it limits, find them.