# Integration Exercise 2

## Imports



In [1]:

%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 [ ]:

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))




In [ ]:

assert True # leave this cell to grade the above integral



### Integral 1

$$I = \int_0^\infty \frac{x^{p-1}dx}{1+x} = \frac{\pi}{sin p \pi}, 0<p<1$$


In [2]:

def integrand(x, p):
return x**(p-1)/(1+x)
def integral_approx(p):
I, e = integrate.quad(integrand, 0, np.inf, args=(p,))
return I
def integral_exact(p):
return np.pi/np.sin(p*np.pi)

print("Numerical: ", integral_approx(.5))
print("Exact    : ", integral_exact(.5))




Numerical:  3.141592653591144
Exact    :  3.14159265359




In [3]:

assert True # leave this cell to grade the above integral



### Integral 2

$$I = \int_0^{2\pi} \frac{dx}{a+bsin(x)} = \frac{2\pi}{\sqrt{a^2+b^2}}$$


In [5]:

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/(a**2+b**2)**.5

print("Numerical: ", integral_approx(.5, .5))
print("Exact    : ", integral_exact(.5, .5))




Numerical:  inf
Exact    :  8.885765876316732




In [6]:

assert True # leave this cell to grade the above integral



### Integral 3

$$I = \int_0^\infty e^{-ax^2}dx = \frac{1}{2}\sqrt{\frac{\pi}{a}}$$


In [7]:

def integrand(x, a):
return np.exp(-a*x**2)
def integral_approx(a):
I, e = integrate.quad(integrand, 0, np.inf, args=(a,))
return I
def integral_exact(a):
return .5*(np.pi/a)**.5

print("Numerical: ", integral_approx(.5))
print("Exact    : ", integral_exact(.5))




Numerical:  1.2533141373154997
Exact    :  1.2533141373155001




In [8]:

assert True # leave this cell to grade the above integral



### Integral 4

$$I = \int_0^1 \frac{ln(x)}{1+x}dx = -\frac{\pi^2}{12}$$


In [11]:

def integrand(x):
return np.log(x)/(1+x)
def integral_approx():
I, e = integrate.quad(integrand, 0, 1)
return I
def integral_exact():
return -np.pi**2/12

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




Numerical:  -0.8224670334241143
Exact    :  -0.8224670334241132




In [12]:

assert True # leave this cell to grade the above integral



### Integral 5

$$I = \int_{-\infty}^\infty \frac{1}{cosh x}dx = \pi$$


In [13]:

def integrand(x):
return 1/np.cosh(x)
def integral_approx():
I, e = integrate.quad(integrand, -np.inf, np.inf)
return I
def integral_exact():
return np.pi

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




Numerical:  3.1415926535897936
Exact    :  3.141592653589793




In [ ]:

assert True # leave this cell to grade the above integral