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Here, we will aim to provide you with a number of exercises and solutions with a varying degree of difficulty. There will also occur exercises that requires you to use functions that has not yet been described in Symbolic mathematics.

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Integration

Exercise 1: Single definite integral

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title	Exercise: Single definite integral

This example is taken from the 2018 exam in TMA4100 - Calculus 1, check out the exam and the solution for hand calculations (only in norwegain). This is task 2.

Compute the intregral

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title	Solution: Single definite integral

Code Block

language	py
title	Solution

from sympy import *

# First we define our variable
x = Symbol('x')
# We then compute the answer to our given interval
ans = integrate(x*log(x**2), (x, 1, 4))     # log() without specifying the base equals ln()
# Finally, we print the answer
print(ans)
print(float(ans))   # Since the answer includes a constant, we have to specify if we want it as a float


# Output:
-15/2 + 8*log(16)	# log() is here the natural logarithm
14.68070977791825

Exercise 2: Double definite integral

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title	Exercise: Double definite integral

Compute the integral

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title	Solution: Double definite integral

Code Block

language	py
title	Solution

from sympy import *

# First we define our variables
x, y = symbols('x y')

# To make our code more readable, we will define the function we will integrate first.
f = x**3*exp(y)

# The integrate function is able to take multiple integrations as parameteres.
# The different integrations are listed according to the integration order: dy -> dx
ans = integrate(f, (y, x**2, 9), (x, 0, 3))

print(ans)
print(float(ans))   # Since the answer includes a constant, we have to specify if we want it as a float


# Output:
-1/2 + 65*exp(9)/4
131674.6138231

Exercise 3: Triple definite integral

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title	Exercise: Triple definite integral

Compute the integral

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title	Solution: Triple definite integral

Code Block

language	py
title	Solution

from sympy import *

# First we define our variables
x, y, z = symbols('x y z')

# The integrate function is able to take multiple integrations as parameteres.
# The different integrations are listed according to the integration order: dz -> dx -> dy.
ans = integrate(1, (z, 24*x, 1), (x, -sqrt(1-y**2), sqrt(1-y**2)), (y, -1, 1))

print(ans)
print(float(ans))   # Since the answer (in this case) is a constant, we have to specify if we want it as a float


# Output:
pi
3.141592653589793

Exercise 4: Single indefinite integral

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title	Exercise: Single indefinite integral

Evaluate

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title	Solution: Single indefinite integral

Code Block

language	py
title	Solution

from sympy import *
 
# First we define our variable
t = Symbol('t')
# We then compute the answer to our given interval
ans = integrate((t**2 - 1)*(4 + 3*t))
# Finally, we print the answer
print(ans)


# Output:
3*t**4/4 + 4*t**3/3 - 3*t**2/2 - 4*t

Remember, when evaluating indefinite integrals, we have to add a general constant to our answer. Python does not include this when integrating, thus we can evaluate the output to as the answer

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Integration

Exercise 1: Single definite integral

Exercise 2: Double definite integral

Exercise 3: Triple definite integral

Exercise 4: Single indefinite integral