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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Integration Exercise 2
Imports
Step1:  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
Step2:  Integral 1
Here is the integral I am performing
Step3:  Integral 2
$$ I_2 = \int_0^\infty \frac{sin^2 px}{x}dx = \frac{\pi p}{2} $$
Step4:  Integral 3
$$ I_3 = \int_0^\infty \frac{1 - cos px}{x}dx = \frac{\pi p}{2} $$
Step5:  Integral 4
$$ I_4 = \int_0^\infty \frac{x}{e^x-1}  dx = \frac{\pi^2}{6} $$
Step6:  Integral 5
$$ I_5 = \int_0^\infty \frac{x}{e^x+1}  dx = \frac{\pi^2}{12} $$
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Python Code:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
from scipy import integrate
import math as m
Explanation: Integration Exercise 2
Imports
End of explanation
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))
assert True # leave this cell to grade the above integral
Explanation: 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:
$$ I_1 = \int_0^\infty \frac{dx}{x^2 + a^2} = \frac{\pi}{2a} $$
End of explanation
# 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))
assert True # leave this cell to grade the above integral
Explanation: Integral 1
Here is the integral I am performing:
$$ I_1 =  \int_0^\frac{\pi}{2} sin^2 x dx = \frac{\pi}{4}  $$
End of explanation
# 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))
assert True # leave this cell to grade the above integral
Explanation: Integral 2
$$ I_2 = \int_0^\infty \frac{sin^2 px}{x}dx = \frac{\pi p}{2} $$
End of explanation
# 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))
assert True # leave this cell to grade the above integral
Explanation: Integral 3
$$ I_3 = \int_0^\infty \frac{1 - cos px}{x}dx = \frac{\pi p}{2} $$
End of explanation
# 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))
assert True # leave this cell to grade the above integral
Explanation: Integral 4
$$ I_4 = \int_0^\infty \frac{x}{e^x-1}  dx = \frac{\pi^2}{6} $$
End of explanation
# 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))
assert True # leave this cell to grade the above integral
Explanation: Integral 5
$$ I_5 = \int_0^\infty \frac{x}{e^x+1}  dx = \frac{\pi^2}{12} $$
End of explanation
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