# 1st: Import the sympy module
import sympy as sp  # Import sympy for symbolic computation

# 2nd: Define the variable and the function
t = sp.symbols('t')  # Define the symbolic variable 't'
f = (2*t + 1) / (t**2 * (t**2 + 1))  # Define the function to integrate

# 3rd: Perform partial fraction decomposition
decomp = sp.apart(f)  # Compute the partial fraction decomposition

# 4th: Compute the integral of the decomposed expression
integral = sp.integrate(decomp, t)  # Perform the integration

# 5th: Output the result
print("The integral is:", integral)  # Print the result of the integral

# 1st: Import the sympy module
import sympy as sp  # Import sympy for symbolic computation

# 2nd: Define the variable and the function
v = sp.symbols('v')  # Define the symbolic variable 'v'
f = (v**4 + 1) / (v**4 - 1)  # Define the function to integrate

# 3rd: Perform partial fraction decomposition
decomp = sp.apart(f)  # Compute the partial fraction decomposition

# 4th: Compute the integral of the decomposed expression
integral = sp.integrate(decomp, v)  # Perform the integration

# 5th: Output the result
print("The integral is:", integral)  # Print the result of the integral

# 1st: Import the sympy module
import sympy as sp  # Import sympy for symbolic computation

# 2nd: Define the variable and the function
s = sp.symbols('s')  # Define the symbolic variable 's'
f = (s**3 + s**2 + s + 3) / ((s**2 + 1)*(s**2 + 3))  # Define the function to integrate

# 3rd: Perform partial fraction decomposition
decomp = sp.apart(f)  # Compute the partial fraction decomposition

# 4th: Compute the integral of the decomposed expression
integral = sp.integrate(decomp, s)  # Perform the integration

# 5th: Output the result
print("The integral is:", integral)  # Print the result of the integral

# 1st: Import the sympy module
import sympy as sp  # Import sympy for symbolic computation

# 2nd: Define the variable
n = sp.symbols('n')  # Define the symbolic variable 'n'

# 3rd: Define the integrand
integrand = (n**3 + 4*n - 2) / (n**3 - 8)  # Define the function to integrate

# 4th: Integrate the expression
integral_result = sp.integrate(integrand, n)  # Perform the integration

# 5th: Display the result
print(integral_result)  # Print the result of the integral

# 1st: Import the sympy module
import sympy as sp  # Import sympy for symbolic computation

# 2nd: Define the variable
m = sp.symbols('m')  # Define the symbolic variable 'm'

# 3rd: Define the integrand
integrand = 1 / ((m**2 + 1)*(m**2 + 4))  # Define the function to integrate

# 4th: Integrate the expression
integral_result = sp.integrate(integrand, m)  # Perform the integration

# 5th: Display the result
print(integral_result)  # Print the result of the integral