5Which substitution is most suitable for evaluating ?
Integration by substitution
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Choosing gives , which appears directly in the integral.
Incorrect! Try again.
6Evaluate .
Integration by substitution
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Let , so . The integral becomes .
Incorrect! Try again.
7Evaluate .
Integration by substitution
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Let , so . Then .
Incorrect! Try again.
8Evaluate .
Integration by substitution
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Let , so . This gives .
Incorrect! Try again.
9Which formula represents integration by parts?
Integration by parts
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Integration by parts follows from the product rule and is given by .
Incorrect! Try again.
10For , which is the usual choice for ?
Integration by parts
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Choosing simplifies it when differentiated, since , while is easy to integrate.
Incorrect! Try again.
11Evaluate .
Integration by parts
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Take and . Then .
Incorrect! Try again.
12Evaluate using integration by parts with and .
Integration by parts
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Here and . Thus, , giving .
Incorrect! Try again.
13Which expression is the correct partial fraction form of ?
Integration by partial fraction
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Combining gives .
Incorrect! Try again.
14Before applying partial fractions to a rational function, what condition should generally be satisfied?
Integration by partial fraction
Easy
A.The denominator contains no variable terms
B.The numerator degree is less than the denominator degree
C.The numerator contains no constant terms
D.The numerator degree equals the denominator degree
Correct Answer: The numerator degree is less than the denominator degree
Explanation:
A rational function should be proper before decomposition. If it is improper, polynomial division is performed first.
Incorrect! Try again.
15Evaluate .
Integration by partial fraction
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Decompose the integrand as and integrate each term.
Incorrect! Try again.
16For the decomposition , what are and ?
Integration by partial fraction
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Combining the fractions gives . Hence and , so and .
Incorrect! Try again.
17What is the value of ?
Properties of definite integrals
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
A definite integral over an interval of zero width is always .
Incorrect! Try again.
18Which identity correctly reverses the limits of integration?
Properties of definite integrals
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
Reversing the limits of a definite integral changes the sign of its value.
Incorrect! Try again.
19If is an even function, which relation is correct?
Properties of definite integrals
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
An even function is symmetric about the -axis, so its integral over is twice its integral over .
Incorrect! Try again.
20If is an odd function, what is ?
Properties of definite integrals
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
For an odd function, the contributions over and cancel, so the integral is .
Incorrect! Try again.
21Evaluate .
General rules of integration
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Apply linearity and the rules , , and .
Incorrect! Try again.
22Find .
General rules of integration
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Since and , the integral is .
Incorrect! Try again.
23A function satisfies and . Which expression represents ?
General rules of integration
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Integrating gives . Using gives , so .
Incorrect! Try again.
24For , evaluate .
General rules of integration
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Increase each exponent by and divide by the new exponent. Thus and .
Incorrect! Try again.
25Evaluate .
Integration by substitution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Let , so . The integral becomes .
Incorrect! Try again.
26Evaluate the definite integral .
Integration by substitution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
With , the limits remain and , and the integral becomes .
Incorrect! Try again.
27Find .
Integration by substitution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Set , so . Then .
Incorrect! Try again.
28Evaluate .
Integration by substitution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Let , giving . The new limits are and , so the integral is .
Incorrect! Try again.
29Evaluate .
Integration by parts
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Using and gives . Thus the integral is .
Incorrect! Try again.
30Which is the correct antiderivative of for ?
Integration by parts
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Take and . Then and , giving .
Incorrect! Try again.
31Evaluate .
Integration by parts
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Integration by parts gives . Evaluating from to yields .
Incorrect! Try again.
32Find .
Integration by parts
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Apply integration by parts twice. First obtain , then use .
Incorrect! Try again.
33Evaluate .
Integration by partial fraction
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The decomposition is . Integrate each term separately.
Incorrect! Try again.
34Find .
Integration by partial fraction
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Since , the fraction decomposes as . Integrating gives the stated logarithms.
Incorrect! Try again.
35Which partial-fraction decomposition is correct for ?
Integration by partial fraction
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Writing and comparing coefficients gives .
Incorrect! Try again.
36Evaluate .
Integration by partial fraction
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The decomposition is . Integrating these terms yields .
Incorrect! Try again.
37Evaluate without directly finding an antiderivative.
Properties of definite integrals
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The function is odd and is even, so their product is odd. The integral of an odd function over is .
Incorrect! Try again.
38Suppose and . Find .
Properties of definite integrals
Medium
A.
B.
C.
D., obtained without averaging the paired terms
Correct Answer:
Explanation:
Pairing with gives . Therefore, the required integral is .
Incorrect! Try again.
39Given and , determine .
Properties of definite integrals
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Additivity gives . Reversing the limits changes the sign, so .
Incorrect! Try again.
40A function satisfies on . Find .
Properties of definite integrals
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The integrals of and over are equal. Thus twice the required integral equals .
Incorrect! Try again.
41Evaluate .
General rules of integration
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Since , the required antiderivative is .
Incorrect! Try again.
42Evaluate .
General rules of integration
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Write and use . The integral becomes .
Incorrect! Try again.
43Find .
General rules of integration
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Split and complete the square: .
Incorrect! Try again.
44Evaluate .
General rules of integration
Hard
A., obtained after replacing by and integrating each resulting term
B.
C.
D.
Correct Answer:
Explanation:
Expanding and using gives , whose integral is the stated result.
Incorrect! Try again.
45Evaluate .
Integration by substitution
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
With , write . Integrating gives the result.
Incorrect! Try again.
46Evaluate .
Integration by substitution
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Using changes the limits to and , giving .
Incorrect! Try again.
47Evaluate the improper integral .
Integration by substitution
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Set . The integral becomes .
Incorrect! Try again.
48For , evaluate .
Integration by substitution
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The substitution gives , leaving .
Incorrect! Try again.
49Evaluate .
Integration by parts
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Applying integration by parts twice reduces the polynomial degree and yields .
Incorrect! Try again.
50For , find .
Integration by parts
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Two applications of integration by parts give .
Incorrect! Try again.
51Evaluate .
Integration by parts
Hard
A., after retaining only the boundary term from integration by parts and omitting the remaining rational integral
B.
C.
D.
Correct Answer:
Explanation:
Integration by parts gives , which simplifies to .
Incorrect! Try again.
52Evaluate .
Integration by parts
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
After integration by parts, divide by . The logarithmic terms cancel, leaving .
Incorrect! Try again.
53Evaluate .
Integration by partial fraction
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The decomposition is . Integrating each term gives the stated expression.
Incorrect! Try again.
54Evaluate .
Integration by partial fraction
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Since , evaluation at the limits gives .
Incorrect! Try again.
55Find .
Integration by partial fraction
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Use and integrate term by term.
Incorrect! Try again.
56Evaluate the improper integral .
Integration by partial fraction
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Partial fractions give . The divergent logarithms cancel in the limiting difference.
Incorrect! Try again.
57Let be integrable on and satisfy . Find .
Properties of definite integrals
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The substitution shows that twice the required integral equals .
Incorrect! Try again.
58If is continuous on , evaluate .
Properties of definite integrals
Hard
A.
B., because the negative half contributes twice the magnitude of the corresponding positive-half integral
C.
D.
Correct Answer:
Explanation:
Because is even and is odd, the integrand is odd. Its integral over the symmetric interval is zero.
Incorrect! Try again.
59Evaluate .
Properties of definite integrals
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Symmetry gives the same integral for . Adding them and using yields .
Incorrect! Try again.
60Let be integrable on and suppose throughout the interval. Evaluate
Properties of definite integrals
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Replacing by produces the complementary fraction. Adding the two equal integral representations gives .
Incorrect! Try again.
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