Integration Exercise 2

Imports



In [14]:

    
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
from scipy import integrate
import math as m

Indefinite integrals

Here is a table of definite integrals. Many of these integrals has a number of parameters $a$, $b$, etc.

Find five of these integrals and perform the following steps:

Typeset the integral using LateX in a Markdown cell.
Define an integrand function that computes the value of the integrand.
Define an integral_approx funciton that uses scipy.integrate.quad to peform the integral.
Define an integral_exact function that computes the exact value of the integral.
Call and print the return value of integral_approx and integral_exact for one set of parameters.

Here is an example to show what your solutions should look like:

Example

Here is the integral I am performing:

$$ I_1 = \int_0^\infty \frac{dx}{x^2 + a^2} = \frac{\pi}{2a} $$



In [15]:

    
def integrand(x, a):
    return 1.0/(x**2 + a**2)

def integral_approx(a):
    # Use the args keyword argument to feed extra arguments to your integrand
    I, e = integrate.quad(integrand, 0, np.inf, args=(a,))
    return I

def integral_exact(a):
    return 0.5*np.pi/a

print("Numerical: ", integral_approx(1.0))
print("Exact    : ", integral_exact(1.0))









    



Numerical:  1.5707963267948966
Exact    :  1.5707963267948966



In [16]:

    
assert True # leave this cell to grade the above integral

Integral 1

Here is the integral I am performing:

$$ I_1 = \int_0^\frac{\pi}{2} sin^2 x dx = \frac{\pi}{4} $$



In [17]:

    
# YOUR CODE HERE
# raise NotImplementedError()
def integrand(x):
    return (np.sin(x))**2

def integral_approx(a):
    I, err = integrate.quad(integrand, 0, np.pi)
    return I

def integral_exact(a):
    return np.pi/4

print("Numerical: ", integral_approx(1.0))
print("Exact    : ", integral_exact(1.0))









    



Numerical:  1.5707963267948966
Exact    :  0.7853981633974483



In [18]:

    
assert True # leave this cell to grade the above integral

Integral 2

$$ I_2 = \int_0^\infty \frac{sin^2 px}{x}dx = \frac{\pi p}{2} $$



In [19]:

    
# YOUR CODE HERE
def integrand(x, p):
    return ((np.sin(p*x))**2)/x

def integral_approx(p):
    # Use the args keyword argument to feed extra arguments to your integrand
    I, e = integrate.quad(integrand, 0, np.inf, args=(p,))
    return I

def integral_exact(p):
    return 0.5*np.pi*p/2

print("Numerical: ", integral_approx(1.0))
print("Exact    : ", integral_exact(1.0))









    



Numerical:  5.066861106505159
Exact    :  0.7853981633974483



In [20]:

    
assert True # leave this cell to grade the above integral

Integral 3

$$ I_3 = \int_0^\infty \frac{1 - cos px}{x}dx = \frac{\pi p}{2} $$



In [21]:

    
# YOUR CODE HERE
def integrand(x, p):
    return (1.0-np.cos(p*x))/x

def integral_approx(p):
    # Use the args keyword argument to feed extra arguments to your integrand
    I, e = integrate.quad(integrand, 0, np.inf, args=(p,))
    return I

def integral_exact(p):
    return 0.5*np.pi*p/2

print("Numerical: ", integral_approx(1.0))
print("Exact    : ", integral_exact(1.0))









    



Numerical:  7.460786724542071
Exact    :  0.7853981633974483



In [22]:

    
assert True # leave this cell to grade the above integral

Integral 4

$$ I_4 = \int_0^\infty \frac{x}{e^x-1} dx = \frac{\pi^2}{6} $$



In [23]:

    
# YOUR CODE HERE
def integrand(x):
    return x/(np.exp(x)-1)

def integral_approx(a):
    I, err = integrate.quad(integrand, 0, np.inf)
    return I

def integral_exact(a):
    return np.pi**2/6

print("Numerical: ", integral_approx(1.0))
print("Exact    : ", integral_exact(1.0))









    



Numerical:  1.6449340668482264
Exact    :  1.6449340668482264



In [24]:

    
assert True # leave this cell to grade the above integral

Integral 5

$$ I_5 = \int_0^\infty \frac{x}{e^x+1} dx = \frac{\pi^2}{12} $$



In [25]:

    
# YOUR CODE HERE
# YOUR CODE HERE
def integrand(x):
    return x/(np.exp(x)+1)

def integral_approx(a):
    I, err = integrate.quad(integrand, 0, np.inf)
    return I

def integral_exact(a):
    return np.pi**2/12

print("Numerical: ", integral_approx(1.0))
print("Exact    : ", integral_exact(1.0))









    



Numerical:  0.822467033424113
Exact    :  0.8224670334241132



In [70]:

    
assert True # leave this cell to grade the above integral