Integration Exercise 2

Imports



In [35]:

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

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 [36]:

    
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 [37]:

    
assert True # leave this cell to grade the above integral

Integral 1

\begin{equation*} I = \int_0^\infty \frac{\sin^2{px}}{x^2}dx = \frac{\pi p}{2} \end{equation*}



In [38]:

    
def integrand1(x,p):
    return (np.sin(p*x)**2)/(x**2)

def integral_approx1(p):
    I1, e1 = integrate.quad(integrand1, 0, np.inf, args=(p,))
    return I1

def integral_exact1(p):
    return np.pi*p/2

print("Numerical: ", integral_approx1(1.0))
print("Exact    : ", integral_exact1(1.0))









    



Numerical:  1.5708678849453777
Exact    :  1.5707963267948966



In [39]:

    
assert True # leave this cell to grade the above integral

Integral 2

\begin{equation*} I = \int_0^\frac{\pi}{2} \sin^2{x} dx = \frac{\pi}{4} \end{equation*}



In [40]:

    
def integrand2(x):
    return (np.sin(x))**2

def integral_approx2():
    I2, e2 = integrate.quad(integrand2, 0, np.pi/2)
    return I2

def integral_exact2():
    return np.pi/4

print("Numerical: ", integral_approx2())
print("Exact    : ", integral_exact2())









    



Numerical:  0.7853981633974483
Exact    :  0.7853981633974483



In [41]:

    
assert True # leave this cell to grade the above integral

Integral 3

\begin{equation*} I = \int_0^\infty e^{-ax} \cos{bx} \; dx = \frac{a}{a^2+b^2} \end{equation*}



In [42]:

    
def integrand3(x,a,b):
    return np.exp(-a*x)*np.cos(b*x)

def integral_approx3(a,b):
    I3, e3 = integrate.quad(integrand3, 0, np.inf, args=(a,b,))
    return I3

def integral_exact3(a,b):
    return a/(a**2+b**2)

print("Numerical: ", integral_approx3(1.0,1.0))
print("Exact    : ", integral_exact3(1.0,1.0))









    



Numerical:  0.5
Exact    :  0.5



In [43]:

    
assert True # leave this cell to grade the above integral

Integral 4

\begin{equation*} I = \int_0^\infty \frac{x}{e^{x}-1}dx = \frac{\pi^2}{6} \end{equation*}



In [44]:

    
def integrand4(x):
    return x/(np.exp(x)-1)

def integral_approx4():
    I4, e4 = integrate.quad(integrand4, 0, np.inf)
    return I4

def integral_exact4():
    return np.pi**2/6

print("Numerical: ", integral_approx4())
print("Exact    : ", integral_exact4())









    



Numerical:  1.6449340668482264
Exact    :  1.6449340668482264



In [45]:

    
assert True # leave this cell to grade the above integral

Integral 5

\begin{equation*} I = \int_0^1 \frac{ln(1+x)}{x}dx = \frac{\pi^2}{12} \end{equation*}



In [46]:

    
def integrand5(x):
    return (np.log(1+x))/(x)

def integral_approx5():
    I5, e5 = integrate.quad(integrand5, 0, 1)
    return I5

def integral_exact5():
    return (np.pi**2)/12

print("Numerical: ", integral_approx5())
print("Exact    : ", integral_exact5())









    



Numerical:  0.8224670334241132
Exact    :  0.8224670334241132



In [47]:

    
assert True # leave this cell to grade the above integral