Integration Exercise 2

Imports



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%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:

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



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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))



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assert True # leave this cell to grade the above integral

Integral 1

YOUR ANSWER HERE



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# YOUR CODE HERE
raise NotImplementedError()



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assert True # leave this cell to grade the above integral

Integral 2

YOUR ANSWER HERE



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# YOUR CODE HERE
raise NotImplementedError()



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assert True # leave this cell to grade the above integral

Integral 3

YOUR ANSWER HERE



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# YOUR CODE HERE
raise NotImplementedError()



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assert True # leave this cell to grade the above integral

Integral 4

YOUR ANSWER HERE



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# YOUR CODE HERE
raise NotImplementedError()



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assert True # leave this cell to grade the above integral

Integral 5

YOUR ANSWER HERE



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# YOUR CODE HERE
raise NotImplementedError()



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assert True # leave this cell to grade the above integral