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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 an integral from the hyperbolic subsection
Step3:  Integral 2
Here is an integral from the exponential functions subsection
Step4:  Integral 3
Here is an integral from the trigonometric functions subsection
Step5:  Integral 4
Here is an integral from the logarithmic functions subsection
Step6:  Integral 5
Here is an integral from the rational and irrational functions subsection
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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
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
def integrand(x,a,b):
    return np.sin(a*x)/np.sinh(b*x)
def integrate_approx(a,b):
    I,e=integrate.quad(integrand,0,np.inf, args=(a,b))
    return I
def integrate_exact(a,b):
    return np.pi/(2*b)*np.tanh(a*np.pi/(2*b))
print('Numerical:', integrate_approx(1.0,2.0))
print('Exact:', integrate_exact(1.0,2.0))
assert True # leave this cell to grade the above integral
Explanation: Integral 1
Here is an integral from the hyperbolic subsection:
\begin{equation}
\int_{0}^{\infty} \frac{\sin ax}{\sinh bx} dx = \frac{\pi}{2b}\tanh \frac{a\pi}{2b}
\end{equation}
End of explanation
def integrand(x,a,b):
    return np.exp(-a*x)*np.cos(b*x)
def integrate_approx(a,b):
    I,e=integrate.quad(integrand,0,np.inf, args=(a,b))
    return I
def integrate_exact(a,b):
    return a/(a**2+b**2)
print('Numerical:', integrate_approx(1.0,2.0))
print('Exact:', integrate_exact(1.0,2.0))
assert True # leave this cell to grade the above integral
Explanation: Integral 2
Here is an integral from the exponential functions subsection:
\begin{equation}
\int_{0}^{\infty} e^{-ax} \cos bx \space dx = \frac{a}{a^{2}+b^{2}}
\end{equation}
End of explanation
def integrand(x,p):
    return (1-np.cos(p*x))/x**2
def integrate_approx(p):
    I,e=integrate.quad(integrand,0,np.inf, args=(p))
    return I
def integrate_exact(p):
    return p*np.pi/2
print('Numerical:', integrate_approx(4.0))
print('Exact:', integrate_exact(4.0))
assert True # leave this cell to grade the above integral
Explanation: Integral 3
Here is an integral from the trigonometric functions subsection:
\begin{equation}
\int_{0}^{\infty} \frac{1-cospx}{x^{2}} dx = \frac{\pi p}{2}
\end{equation}
End of explanation
def integrand(x,a,b):
    return np.log(a**2+x**2)/(b**2+x**2)
def integrate_approx(a,b):
    I,e=integrate.quad(integrand,0,np.inf, args=(a,b))
    return I
def integrate_exact(a,b):
    return np.pi/b*np.log(a+b)
print('Numerical:', integrate_approx(3.0,4.0))
print('Exact:', integrate_exact(3.0,4.0))
assert True # leave this cell to grade the above integral
Explanation: Integral 4
Here is an integral from the logarithmic functions subsection:
\begin{equation}
\int_{0}^{\infty} \frac{\ln (a^{2}+x^{2})}{b^{2}+x^{2}} dx = \frac{\pi}{b}ln(a+b) \space \space a,b>0
\end{equation}
End of explanation
def integrand(x,a,b):
    return np.sqrt(a**2-x**2)
def integrate_approx(a,b):
    I,e=integrate.quad(integrand,0,a, args=(a,b))
    return I
def integrate_exact(a,b):
    return np.pi*a**2/4
print('Numerical:', integrate_approx(1.0,2.0))
print('Exact:', integrate_exact(1.0,2.0))
assert True # leave this cell to grade the above integral
Explanation: Integral 5
Here is an integral from the rational and irrational functions subsection:
\begin{equation}
\int_{0}^{a} \sqrt{a^{2}-x^{2}} dx = \frac{\pi a^{2}}{4}
\end{equation}
End of explanation
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