2What is the indefinite integral of a constant with respect to ?
general rules of integration
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Since the derivative of is , .
Incorrect! Try again.
3Evaluate .
general rules of integration
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Integrate each term separately: and .
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4What is for ?
general rules of integration
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The standard logarithmic rule is .
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5Evaluate .
integration by substitution
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Let , so . The integral becomes .
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6Which substitution is most suitable for ?
integration by substitution
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The denominator is , and its derivative is proportional to the numerator.
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7Evaluate .
integration by substitution
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Using introduces the factor , giving .
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8Evaluate .
integration by substitution
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Let , so . Then integrate .
Incorrect! Try again.
9Which formula represents integration by parts?
integration by parts
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Integration by parts follows directly from rearranging the product rule for differentiation.
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10Evaluate .
integration by parts
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Choose and . Then and .
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11For , which choice of is generally most convenient?
integration by parts
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Choosing is convenient because its derivative simplifies to .
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12Evaluate .
integration by parts
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Take and . Then , giving .
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13When is a rational function called a proper fraction?
integration by partial fraction
Easy
A.When the numerator degree exceeds the denominator degree
B.When the numerator degree equals the denominator degree
C.When the denominator is a constant polynomial
D.When the numerator degree is less than the denominator degree
Correct Answer: When the numerator degree is less than the denominator degree
Explanation:
A rational function is proper when the degree of its numerator is smaller than the degree of its denominator.
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14Which is the correct partial fraction decomposition of ?
integration by partial fraction
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Combining the correct fractions gives .
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15Which partial fraction form is appropriate for ?
integration by partial fraction
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Each distinct linear factor in the denominator receives a constant numerator.
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16Find the partial fraction decomposition of .
integration by partial fraction
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Combining the fractions gives .
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17Evaluate .
properties of definite integrals
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
An antiderivative is . Thus .
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18How are and related?
properties of definite integrals
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Reversing the limits of a definite integral changes its sign.
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19What is the value of ?
properties of definite integrals
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
A definite integral with identical lower and upper limits is always zero.
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20If is an odd function, what is ?
properties of definite integrals
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
For an odd function, the signed areas over and cancel.
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21Evaluate .
general rules of integration
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Integrate each term separately using the power, logarithmic, and exponential integration rules.
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22Find .
general rules of integration
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Expand the integrand as and apply the power rule to both terms.
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23Evaluate .
general rules of integration
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Use and .
Incorrect! Try again.
24Determine .
general rules of integration
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Rewrite the integrand as , then integrate using the power rule.
Incorrect! Try again.
25Evaluate .
integration by substitution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Let , so . Then integrate .
Incorrect! Try again.
26Calculate .
integration by substitution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
With , , and the transformed limits remain and .
Incorrect! Try again.
27Find .
integration by substitution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Set , giving and the integral .
Incorrect! Try again.
28Evaluate .
integration by substitution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Let , so , and integrate .
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29Evaluate for .
integration by parts
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Choose and , then apply .
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30Find .
integration by parts
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Take and , so , and apply integration by parts.
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31Calculate .
integration by parts
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Integration by parts gives .
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32Evaluate .
integration by parts
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Applying integration by parts twice and solving for the original integral yields the stated result.
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33Find .
integration by partial fraction
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Decompose as and integrate both terms.
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34Evaluate .
integration by partial fraction
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Factor the denominator and decompose the integrand as .
Incorrect! Try again.
35Find .
integration by partial fraction
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The decomposition is , which can be integrated directly.
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36Evaluate .
integration by partial fraction
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Decompose the integrand as and integrate.
Incorrect! Try again.
37If and , find .
properties of definite integrals
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
By linearity, the value is .
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38Suppose is an odd continuous function. What is if ?
properties of definite integrals
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The odd part integrates to zero over , while the even part contributes .
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39Given and , determine .
properties of definite integrals
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The correct option follows directly from the given concept and definitions.
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40A continuous function satisfies for . Find .
properties of definite integrals
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Using the substitution , the integral equals .
Incorrect! Try again.
41Evaluate .
general rules of integration
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Polynomial division gives . Integrating termwise yields the stated result.
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42A twice-differentiable function satisfies , , and . Determine .
general rules of integration
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Integrating twice gives and after applying both initial conditions.
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43Evaluate on an interval where is defined.
general rules of integration
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Use and integrate after rewriting .
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44For differentiable functions and , with , evaluate .
general rules of integration
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The integrand is exactly the quotient-rule derivative .
Incorrect! Try again.
45Evaluate .
integration by substitution
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Set . Then , reducing the integral to .
Incorrect! Try again.
46For , evaluate .
integration by substitution
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
With , the integral becomes , whose value for is .
Incorrect! Try again.
47Evaluate .
integration by substitution
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Let , so . The resulting integral is .
Incorrect! Try again.
48For , evaluate .
integration by substitution
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Set , giving . Apply the standard antiderivative for a semicircular integrand.
Incorrect! Try again.
49For , evaluate .
integration by parts
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Choose and . Then and , leaving the integral of .
Incorrect! Try again.
50Evaluate .
integration by parts
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Applying integration by parts twice gives the antiderivative . Evaluation at and gives the result.
Incorrect! Try again.
51Evaluate .
integration by parts
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Take and . The remaining integrand simplifies using .
Incorrect! Try again.
52For , evaluate .
integration by parts
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Two applications of integration by parts reduce the powers of and produce .
Incorrect! Try again.
53Evaluate .
integration by partial fraction
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The decomposition is . Integrating each term gives the result.
Incorrect! Try again.
54Evaluate .
integration by partial fraction
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Decompose the integrand as , then integrate the two logarithmic derivatives.
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55Evaluate the improper integral .
integration by partial fraction
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Since , the logarithmic terms cancel at infinity and leave .
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56Evaluate on an interval not containing .
integration by partial fraction
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The decomposition is . Integrating gives the displayed expression.
Incorrect! Try again.
57An integrable function satisfies for . Find .
properties of definite integrals
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Reflection gives . Thus twice the required integral equals .
Incorrect! Try again.
58Evaluate .
properties of definite integrals
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The first term is odd and integrates to zero. For the even term, use and symmetry.
Incorrect! Try again.
59A function has period and satisfies . Find .
properties of definite integrals
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The interval has length , exactly four periods. Every interval of length has integral , so the result is .
Incorrect! Try again.
60Using symmetry and transformations of definite integrals, evaluate .
properties of definite integrals
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Reflection shows the sine and cosine logarithmic integrals are equal. Adding them and using produces .
Incorrect! Try again.
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