# Integration Exercise 2

## Imports



In [67]:

%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 [68]:

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 [69]:

assert True # leave this cell to grade the above integral



### Integral 1

$$I_1 = \int_0^\frac{\pi}{2} \sin^2{x} {dx} = \frac{\pi}{4}$$


In [70]:

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

def integral_approx(a):
# Use the args keyword argument to feed extra arguments to your integrand
I, e = integrate.quad(integrand, 0, np.pi/2, args=(a,))
return I

def integral_exact(a):
return np.pi/4

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




Numerical:  0.7853981633974483
Exact    :  0.7853981633974483




In [71]:

assert True # leave this cell to grade the above integral



### Integral 2

$$I_2 = \int_0^\infty \frac{x\sin{mx}}{x^2 + a^2} {dx} = \frac{\pi}{2} e^{-ma}$$


In [127]:

def integrand(x, m, a):
return (x*np.sin((m*x)))/(x**2 + a**2)

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

def integral_exact(m, a):
return ((np.pi)*.5) *(np.exp(-1*m*a))

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




Numerical:  1.2549750860426012
Exact    :  0.577863674895




In [73]:

assert True # leave this cell to grade the above integral



### Integral 3

$$I_3 = \int_0^\frac{\pi}{2} \sin{ax^2} {dx} = \frac{1}{2} \sqrt{\frac{\pi}{2 \pi}}$$


In [120]:

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

def integral_approx(a):
# Use the args keyword argument to feed extra arguments to your integrand
I, e = integrate.quad(integrand, 0, (np.pi/2), args=(a))
return I

def integral_exact(a):
return .5*(np.sqrt((np.pi)/np.pi*2*np.pi))

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




Numerical:  0.9999999999999999
Exact    :  1.25331413732




In [ ]:

assert True # leave this cell to grade the above integral



### Integral 4

$$I_4 = \int_0^\infty e^{-ax} \cos{bx}{dx}= \frac{a}{a^2+b^2}$$


In [125]:

def integrand(a, x, b):
return np.exp(-a*x) * np.cos(b*x)

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

def integral_exact(a, b):
return a / (a**2 + b**2)

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




Numerical:  0.5403023058681399
Exact    :  0.5




In [ ]:

assert True # leave this cell to grade the above integral



### Integral 5

$$I_5 = \int_0^\infty e^{-ax^{2}} {dx}= \frac{1}{2} \sqrt{\frac{\pi}{a}}$$


In [101]:

def integrand(x, a):
return np.exp((-a)*(x**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.sqrt((np.pi)/a))

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




Numerical:  0.8862269254527579
Exact    :  0.886226925453




In [ ]:

assert True # leave this cell to grade the above integral