Generating tables of derivatives and integrals

Description

This example shows how SymPy can be used to automatically generate tables of, for instance, derivatives and integrals. We print the output in TeX (replacing the \$-signs in the TeX output with \ [ \ ]-tags and subtracting \operatorname commands for display on the wiki).

from sympy import *

def derivative_table(functions, x):
for f in functions:
s = printing.latex(Eq(Derivative(f, x), diff(f, x)))
print ":$" + s + "$", "\n"

def integral_table(functions, x):
for f in functions:
s = printing.latex(Eq(Integral(f,x), integrate(f, x)))
print ":$" + s + "$", "\n"

var('x')

print "### Derivatives"
derivative_table([cos(x)/(1 + sin(x)**i) for i in range(1, 5)], x)

print "### Integrals"
integral_table([x**i * exp(i*x) for i in range(1, 5)], x)

Output

Derivatives

[ \frac{\partial}{\partial x}\left(\frac{\cos\left(x\right)}{1 + \sin\left(x\right)}\right) = - \frac{\sin\left(x\right)}{1 + \sin\left(x\right)} - \frac{\cos^{2}\left(x\right)}{\left(1 + \sin\left(x\right)\right)^{2}} ]

[ \frac{\partial}{\partial x}\left(\frac{\cos\left(x\right)}{1 + \sin^{2}\left(x\right)}\right) = - \frac{\sin\left(x\right)}{1 + \sin^{2}\left(x\right)} - 2 \frac{\cos^{2}\left(x\right) \sin\left(x\right)}{\left(1 + \sin^{2}\left(x\right)\right)^{2}} ]

[ \frac{\partial}{\partial x}\left(\frac{\cos\left(x\right)}{1 + \sin^{3}\left(x\right)}\right) = - \frac{\sin\left(x\right)}{1 + \sin^{3}\left(x\right)} - 3 \frac{\cos^{2}\left(x\right) \sin^{2}\left(x\right)}{\left(1 + \sin^{3}\left(x\right)\right)^{2}} ]

[ \frac{\partial}{\partial x}\left(\frac{\cos\left(x\right)}{1 + \sin^{4}\left(x\right)}\right) = - \frac{\sin\left(x\right)}{1 + \sin^{4}\left(x\right)} - 4 \frac{\cos^{2}\left(x\right) \sin^{3}\left(x\right)}{\left(1 + \sin^{4}\left(x\right)\right)^{2}} ]

Integrals

[ \int x {e}^{x},dx = - {e}^{x} + x {e}^{x} ]

[ \int {x}^{2} {e}^{2 x},dx = \frac{1}{4} {e}^{2 x} + \frac{1}{2} {x}^{2}{e}^{2 x} - \frac{1}{2} x {e}^{2 x} ]

[ \int {x}^{3} {e}^{3 x},dx = - \frac{2}{27} {e}^{3 x} - \frac{1}{3} {x}^{2} {e}^{3 x} + \frac{1}{3} {x}^{3} {e}^{3 x} + \frac{2}{9} x {e}^{3 x} ]

[ \int {x}^{4} {e}^{4 x},dx = \frac{3}{128} {e}^{4 x} - \frac{3}{32} x {e}^{4 x} - \frac{1}{4} {x}^{3} {e}^{4 x} + \frac{1}{4} {x}^{4} {e}^{4 x} + \frac{3}{16} {x}^{2} {e}^{4 x} ]