1
What is the indefinite integral of a constant with respect to ?
Correct Answer:
Explanation:
The integral of a constant is , plus the constant of integration , because the derivative of is .
2
Evaluate .
Correct Answer:
Explanation:
Using the power rule for integration , where .
3
What does the 'C' represent in an indefinite integral?
A. The upper limit of integration
B. The constant of integration
C. The derivative of the function
D. The area under the curve
Correct Answer: The constant of integration
Explanation:
Because the derivative of a constant is zero, there are infinitely many antiderivatives for a function, differing only by a constant value .
4
Calculate .
Correct Answer:
Explanation:
The exponential function is unique because it is its own derivative and antiderivative.
5
Evaluate .
Correct Answer:
Explanation:
The power rule fails for . The antiderivative of is the natural logarithm .
6
What is ?
Correct Answer:
Explanation:
The derivative of is , so the antiderivative of is .
7
What is ?
8
Evaluate .
Correct Answer:
Explanation:
Integrate each term separately: and .
9
Find .
Correct Answer:
Explanation:
We know from differential calculus that .
10
Evaluate .
Correct Answer:
Explanation:
Rewrite as . Using the power rule: .
11
What property allows us to write ?
A. Sum Rule
B. Chain Rule
C. Power Rule
D. Product Rule
Correct Answer: Sum Rule
Explanation:
The Sum Rule states that the integral of a sum is the sum of the integrals.
12
Evaluate .
Correct Answer:
Explanation:
Using the constant multiple and power rules: .
13
If is the antiderivative of , then is equal to:
Correct Answer:
Explanation:
This is the Fundamental Theorem of Calculus (Part 2).
14
Evaluate the definite integral .
Correct Answer: 8
Explanation:
Antiderivative is . Evaluated from 0 to 2: .
15
What is the value of ?
Correct Answer: 0
Explanation:
An integral where the upper and lower limits are the same is always 0, representing an area with zero width.
16
If and , what is ?
Correct Answer: 6
Explanation:
Using the property of intervals: . Therefore, , so the result is .
17
Evaluate .
Correct Answer:
Explanation:
Antiderivative is . Evaluated: .
18
Which relation is true regarding switching the limits of integration?
Correct Answer:
Explanation:
Reversing the limits of integration changes the sign of the result.
19
Evaluate .
Correct Answer: 2
Explanation:
Antiderivative of is . Evaluated: .
20
If is an odd function, then equals:
Correct Answer: $0$
Explanation:
For an odd function, the signed area on the negative side cancels out the signed area on the positive side exactly.
21
Calculate .
Correct Answer:
Explanation:
Antiderivative is . Evaluated: .
22
What is the primary purpose of Integration by Substitution?
A. To find the area between two curves
B. To reverse the Chain Rule
C. To integrate products of functions
D. To integrate rational functions
Correct Answer: To reverse the Chain Rule
Explanation:
Integration by substitution is used to simplify integrals that suggest the result of a chain rule derivative (composite functions).
23
To solve , which substitution is best?
Correct Answer:
Explanation:
If , then , which perfectly matches the remaining terms in the integral.
24
Solve using substitution.
Correct Answer:
Explanation:
Let , so or . The integral becomes .
25
Evaluate .
Correct Answer:
Explanation:
Using substitution , . We divide by the derivative of the inside function, resulting in the factor .
26
Determine .
Correct Answer:
Explanation:
Let , then . The integral becomes .
27
Solve .
Correct Answer:
Explanation:
Let , then . Integral is .
28
Which of the following requires Integration by Parts?
Correct Answer:
Explanation:
is a product of two unrelated functions where standard substitution does not work. can be solved by simple substitution ().
29
What is the formula for Integration by Parts?
Correct Answer:
Explanation:
This formula is derived from the Product Rule for differentiation.
30
In the LIATE rule for choosing 'u' in integration by parts, what does 'L' stand for?
A. Linear
B. Long
C. Limit
D. Logarithmic
Correct Answer: Logarithmic
Explanation:
LIATE stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. It suggests the priority for choosing .
31
Apply integration by parts to . If , what is ?
Correct Answer:
Explanation:
In integration by parts , if we select , the remaining part of the integrand is .
32
Using integration by parts on , the result is:
Correct Answer:
Explanation:
Let . Then . Formula gives .
33
To integrate using parts, we choose and . What is ?
Correct Answer:
Explanation:
Since , integrating gives .
34
What is the result of ?
35
Evaluate using substitution.
36
The geometric interpretation of for is:
A. The volume of rotation
B. The area under the curve from to
C. The length of the curve
D. The slope of the tangent line
Correct Answer: The area under the curve from to
Explanation:
The definite integral represents the net signed area bounded by the graph of the function and the x-axis.
37
Find the area under the curve from to .
38
Calculate the area under between and .
39
Find the area under the constant line from to .
Correct Answer: 25
Explanation:
Geometric area is a rectangle with height 5 and width . Area . Or .
40
Find the area under from to .
41
What is the area under the curve from to ?
42
Evaluate to find the area under the cosine curve.
43
If calculating the area between a curve and the x-axis, and the curve dips below the x-axis, the integral result for that section is:
A. Zero
B. Negative
C. Undefined
D. Positive
Correct Answer: Negative
Explanation:
Definite integrals treat area below the x-axis as negative signed area.
44
Find the area bounded by , the x-axis, and .
45
Evaluate .
Correct Answer:
Explanation:
Sub , . Need factor 1/3. Integral .
46
What is ?
Correct Answer:
Explanation:
Rewrite as . Let , . Result .
47
When changing variables in a definite integral using substitution , what must happen to the limits of integration?
A. They become 0 and 1
B. They must be transformed to and
C. They switch places
D. They remain the same
Correct Answer: They must be transformed to and
Explanation:
The limits must correspond to the variable of integration. If changing from to , limits change from to .
48
Integration is often described as the reverse process of:
A. Multiplication
B. Exponentiation
C. Factorization
D. Differentiation
Correct Answer: Differentiation
Explanation:
Integration finds the antiderivative, reversing the operation of differentiation.
49
Find .
50
Which rule helps solve ?
A. Sum Rule
B. Power Rule
C. Substitution Rule
D. Integration by Parts (applied twice)
Correct Answer: Integration by Parts (applied twice)
Explanation:
Since there is an term multiplied by a trig function, you must apply integration by parts twice to reduce the power of to 0.