%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
"""


import math

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

def integral_approx(a):
    I, err = integrate.quad(integrand, 0, a, args=(a,))
    return I

def integral_exact(a):
    return math.pi*a**2/4

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

assert True # leave this cell to grade the above integral

"""
Explanation: Integral 1
\begin{equation}
I_1 = \int_0^{a} {\sqrt{a^2 - x^2}} dx= \frac {\pi a^2}{4}
\end{equation}
End of explanation
"""


def integrand(x):
    return math.log(x) / (1+x)

def integral_approx():
    I, err = integrate.quad(integrand, 0, 1)
    return I

def integral_exact():
    return -(math.pi**2)/12

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


assert True # leave this cell to grade the above integral

"""
Explanation: Integral 2
\begin{equation}
I_2 = \int_0^1 \frac {\ln{x}}{1+x} dx = - \frac {\pi^2}{12}
\end{equation}
End of explanation
"""


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

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

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

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

assert True # leave this cell to grade the above integral

"""
Explanation: Integral 3
\begin{equation}
I_3 = \int_0^{\infty} \frac {x}{e^x -1} dx = \frac {\pi^2}{6}
\end{equation}
End of explanation
"""


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

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

def integral_exact(a):
    return 1/2 * math.sqrt(math.pi/a)

print("Numerical: ", integral_approx(1))
print("Exact: ", integral_exact(1))

assert True # leave this cell to grade the above integral

"""
Explanation: Integral 4
\begin{equation}
I_4 = \int_0^\infty e^{-ax^2} dx = \frac {1}{2} \sqrt{\frac{\pi}{a}}
\end{equation}
End of explanation
"""


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

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

def integral_exact():
    return math.pi/4

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

assert True # leave this cell to grade the above integral

"""
Explanation: Integral 5
\begin{equation}
I_5 = \int_0^{\frac{\pi}{2}} \sin^2{x} \: dx = \frac {\pi}{4}
\end{equation}
End of explanation
"""


