# Integration Exercise 2

## Imports



In [2]:

%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:

1. Typeset the integral using LateX in a Markdown cell.
2. Define an integrand function that computes the value of the integrand.
3. Define an integral_approx funciton that uses scipy.integrate.quad to peform the integral.
4. Define an integral_exact function that computes the exact value of the integral.
5. 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 [7]:

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 [8]:

assert True # leave this cell to grade the above integral



### Integral 1

$$I_1 = \int_0^a {\sqrt{a^2-x^2} dx} = \frac{\pi a^2}{4}$$


In [22]:

def integrand(x, a):
return np.sqrt(a**2 - x**2)

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

def integral_exact(a):
return 0.25*np.pi

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




Numerical:  0.7853981633974481
Exact    :  0.7853981633974483




In [23]:

assert True # leave this cell to grade the above integral



### Integral 2

$$I_2 = \int_0^{\frac{\pi}{2}} {\sin^2{x}}{ } {dx} = \frac{\pi}{4}$$


In [26]:

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

def integral_approx():
I, e = integrate.quad(integrand, 0, np.pi/2)
return I

def integral_exact():
return 0.25*np.pi

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




Numerical:  0.7853981633974483
Exact    :  0.7853981633974483




In [ ]:

assert True # leave this cell to grade the above integral



### Integral 3

$$I_3 = \int_0^{2\pi} \frac{dx}{a+b\sin{x}} = {\frac{2\pi}{\sqrt{a^2-b^2}}}$$


In [66]:

def integrand(x,a,b):
return 1/(a+ b*np.sin(x))

def integral_approx(a,b):
I, e = integrate.quad(integrand, 0, 2*np.pi,args=(a,b))
return I

def integral_exact(a,b):
return 2*np.pi/np.sqrt(a**2-b**2)

print("Numerical: ", integral_approx(10,0))
print("Exact    : ", integral_exact(10,0))




Numerical:  0.6283185307179587
Exact    :  0.628318530718




In [ ]:

assert True # leave this cell to grade the above integral



### Integral 4

$$I_4 = \int_0^{\infty} \frac{x}{e^{x}+1} = {\frac{\pi^2}{12}}$$


In [72]:

def integrand(x):
return x/(np.exp(x)+1)

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

def integral_exact():
return (1/12)*np.pi**2

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




Numerical:  0.822467033424113
Exact    :  0.8224670334241131




In [73]:

assert True # leave this cell to grade the above integral



### Integral 5

$$I_5 = \int_0^{\infty} \frac{x}{e^{x}-1} = {\frac{\pi^2}{6}}$$


In [74]:

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

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

def integral_exact():
return (1/6)*np.pi**2

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




Numerical:  1.6449340668482264
Exact    :  1.6449340668482262




In [ ]:

assert True # leave this cell to grade the above integral