Generating tables of derivatives and integrals

From SymPy

Contents 1 Description 2 Output 2.1 Derivatives 2.2 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 <math></math>-tags for display on the wiki).

<source lang=python> from sympy import *

def derivative_table(functions, x):

   for f in functions:
       s = printing.latex(Eq(Derivative(f, x), diff(f, x)))
       print ":<math>" + s[1:-1] + "</math>", "\n"

def integral_table(functions, x):

   for f in functions:
       s = printing.latex(Eq(Integral(f,x), integrate(f, x)))
       print ":<math>" + s[1:-1] + "</math>", "\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) </source>

Output

Derivatives

<math>\frac{\partial}{\partial x}\left(\frac{\operatorname{cos}\left(x\right)}{1 + \operatorname{sin}\left(x\right)}\right) = - \frac{\operatorname{sin}\left(x\right)}{1 + \operatorname{sin}\left(x\right)} - \frac{\operatorname{cos}^{2}\left(x\right)}{\left(1 + \operatorname{sin}\left(x\right)\right)^{2}}</math>

<math>\frac{\partial}{\partial x}\left(\frac{\operatorname{cos}\left(x\right)}{1 + \operatorname{sin}^{2}\left(x\right)}\right) = - \frac{\operatorname{sin}\left(x\right)}{1 + \operatorname{sin}^{2}\left(x\right)} - 2 \frac{\operatorname{c

os}^{2}\left(x\right) \operatorname{sin}\left(x\right)}{\left(1 + \operatorname{sin}^{2}\left(x\right)\right)^{2}}</math>

<math>\frac{\partial}{\partial x}\left(\frac{\operatorname{cos}\left(x\right)}{1 + \operatorname{sin}^{3}\left(x\right)}\right) = - \frac{\operatorname{sin}\left(x\right)}{1 + \operatorname{sin}^{3}\left(x\right)} - 3 \frac{\operatorname{c

os}^{2}\left(x\right) \operatorname{sin}^{2}\left(x\right)}{\left(1 + \operatorname{sin}^{3}\left(x\right)\right)^{2}}</math>

<math>\frac{\partial}{\partial x}\left(\frac{\operatorname{cos}\left(x\right)}{1 + \operatorname{sin}^{4}\left(x\right)}\right) = - \frac{\operatorname{sin}\left(x\right)}{1 + \operatorname{sin}^{4}\left(x\right)} - 4 \frac{\operatorname{c

os}^{2}\left(x\right) \operatorname{sin}^{3}\left(x\right)}{\left(1 + \operatorname{sin}^{4}\left(x\right)\right)^{2}}</math>

Integrals

<math>\int x {e}^{x}\,dx = - {e}^{x} + x {e}^{x}</math>

<math>\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}</math>

<math>\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}</math>

<math>\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}</math>