Integration
Q. Evaluate the integral ∫ (3x^2 + 2x) dx. (2020)
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A.
x^3 + x^2 + C
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B.
x^3 + x^2 + 2C
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C.
x^3 + x^2 + 1
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D.
x^3 + 2x + C
Solution
The integral is (3/3)x^3 + (2/2)x^2 + C = x^3 + x^2 + C.
Correct Answer: A — x^3 + x^2 + C
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Q. Evaluate the integral ∫(3x^2 + 2)dx. (2022)
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A.
x^3 + 2x + C
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B.
x^3 + 2x^2 + C
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C.
x^3 + 2x^3 + C
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D.
3x^3 + 2x + C
Solution
Integrating term by term, ∫3x^2dx = x^3 and ∫2dx = 2x. Thus, ∫(3x^2 + 2)dx = x^3 + 2x + C.
Correct Answer: A — x^3 + 2x + C
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Q. Evaluate ∫ (2x + 3) dx. (2022)
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A.
x^2 + 3x + C
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B.
x^2 + 3 + C
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C.
x^2 + 3x + 1
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D.
2x^2 + 3 + C
Solution
The integral is (2/2)x^2 + 3x + C = x^2 + 3x + C.
Correct Answer: A — x^2 + 3x + C
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Q. Evaluate ∫ (4x^3 - 2x) dx. (2019)
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A.
x^4 - x^2 + C
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B.
x^4 - x^2 + 2C
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C.
x^4 - x + C
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D.
4x^4 - 2x^2 + C
Solution
The integral is (4/4)x^4 - (2/2)x^2 + C = x^4 - x^2 + C.
Correct Answer: A — x^4 - x^2 + C
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Q. Evaluate ∫ (5 - 3x) dx. (2022)
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A.
5x - (3/2)x^2 + C
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B.
5x - (3/3)x^2 + C
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C.
5x - (3/4)x^2 + C
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D.
5x - (3/5)x^2 + C
Solution
The integral is 5x - (3/2)x^2 + C.
Correct Answer: A — 5x - (3/2)x^2 + C
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Q. Evaluate ∫(2x^2 + 3x + 1)dx. (2021)
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A.
(2/3)x^3 + (3/2)x^2 + x + C
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B.
(2/3)x^3 + (3/2)x + C
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C.
(2/3)x^3 + (3/2)x^2 + C
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D.
(2/3)x^3 + 3x + C
Solution
Integrating term by term: ∫2x^2dx = (2/3)x^3, ∫3xdx = (3/2)x^2, and ∫1dx = x. Thus, ∫(2x^2 + 3x + 1)dx = (2/3)x^3 + (3/2)x^2 + x + C.
Correct Answer: A — (2/3)x^3 + (3/2)x^2 + x + C
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Q. Evaluate ∫(5x^4)dx. (2020)
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A.
(5/5)x^5 + C
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B.
(1/5)x^5 + C
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C.
(5/4)x^4 + C
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D.
(1/4)x^4 + C
Solution
The integral of 5x^4 is (5/5)x^5 + C = x^5 + C.
Correct Answer: A — (5/5)x^5 + C
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Q. Find the integral of (2x + 1)^3 dx. (2019)
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A.
(1/4)(2x + 1)^4 + C
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B.
(1/3)(2x + 1)^4 + C
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C.
(1/5)(2x + 1)^4 + C
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D.
(1/2)(2x + 1)^4 + C
Solution
Using substitution, the integral is (1/4)(2x + 1)^4 + C.
Correct Answer: A — (1/4)(2x + 1)^4 + C
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Q. Find the integral of (2x + 3)dx. (2022)
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A.
x^2 + 3x + C
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B.
x^2 + 3x + 1
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C.
x^2 + 3 + C
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D.
2x^2 + 3x + C
Solution
Integrating term by term: ∫2xdx = x^2 and ∫3dx = 3x. Thus, ∫(2x + 3)dx = x^2 + 3x + C.
Correct Answer: A — x^2 + 3x + C
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Q. Find the integral of cos(2x)dx. (2023)
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A.
(1/2)sin(2x) + C
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B.
sin(2x) + C
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C.
(1/2)cos(2x) + C
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D.
2sin(2x) + C
Solution
The integral of cos(kx) is (1/k)sin(kx) + C. Here, k=2, so the integral is (1/2)sin(2x) + C.
Correct Answer: A — (1/2)sin(2x) + C
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Q. Find the integral of cos(x). (2023)
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A.
sin(x) + C
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B.
-sin(x) + C
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C.
cos(x) + C
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D.
-cos(x) + C
Solution
The integral of cos(x) is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. Find the integral of cos(x)dx. (2023)
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A.
sin(x) + C
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B.
-sin(x) + C
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C.
cos(x) + C
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D.
-cos(x) + C
Solution
The integral of cos(x) is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. Find the integral of e^x dx. (2022)
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A.
e^x + C
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B.
e^x
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C.
x e^x + C
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D.
ln(e^x) + C
Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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Q. Find the integral of sin(x). (2020)
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A.
-cos(x) + C
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B.
cos(x) + C
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C.
sin(x) + C
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D.
-sin(x) + C
Solution
The integral of sin(x) is -cos(x) + C.
Correct Answer: A — -cos(x) + C
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Q. Find the integral of sin(x)dx. (2020)
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A.
-cos(x) + C
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B.
cos(x) + C
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C.
sin(x) + C
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D.
-sin(x) + C
Solution
The integral of sin(x) is -cos(x) + C.
Correct Answer: A — -cos(x) + C
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Q. Find the integral of x^5 dx. (2020)
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A.
(1/6)x^6 + C
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B.
(1/5)x^6 + C
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C.
(1/4)x^6 + C
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D.
(1/7)x^6 + C
Solution
The integral is (1/6)x^6 + C.
Correct Answer: B — (1/5)x^6 + C
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Q. Find ∫ (5x^4) dx. (2020)
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A.
x^5 + C
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B.
x^5 + 5C
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C.
x^5 + 1
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D.
5x^5 + C
Solution
The integral is (5/5)x^5 + C = x^5 + C.
Correct Answer: A — x^5 + C
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Q. Find ∫ (6x^2 - 4) dx. (2019)
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A.
2x^3 - 4x + C
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B.
2x^3 - 2x + C
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C.
2x^3 - 4 + C
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D.
3x^3 - 4x + C
Solution
The integral is (6/3)x^3 - 4x + C = 2x^3 - 4x + C.
Correct Answer: A — 2x^3 - 4x + C
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-
A.
e^x + C
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B.
e^x
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C.
x e^x + C
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D.
ln(e^x) + C
Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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Q. What is the integral of 1/x? (2022)
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A.
ln
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B.
x
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C.
+ C
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D.
1/x + C
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.
x + C
-
.
x^2 + C
Solution
The integral of 1/x is ln|x| + C.
Correct Answer: A — ln
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Q. What is the integral of cos(2x) dx? (2021)
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A.
(1/2)sin(2x) + C
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B.
sin(2x) + C
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C.
(1/2)cos(2x) + C
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D.
2sin(2x) + C
Solution
The integral of cos(2x) is (1/2)sin(2x) + C.
Correct Answer: A — (1/2)sin(2x) + C
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Q. What is the integral of e^x with respect to x? (2023)
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A.
e^x + C
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B.
xe^x + C
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C.
e^x/x + C
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D.
x^2e^x + C
Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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Q. What is the integral of e^x? (2023)
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A.
e^x + C
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B.
xe^x + C
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C.
e^x/x + C
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D.
x^2e^x + C
Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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Q. What is the integral of sin(x) dx? (2021)
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A.
-cos(x) + C
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B.
cos(x) + C
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C.
sin(x) + C
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D.
-sin(x) + C
Solution
The integral of sin(x) is -cos(x) + C.
Correct Answer: A — -cos(x) + C
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Q. What is the integral of tan(x) dx? (2023)
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A.
-ln
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B.
cos(x)
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C.
+ C
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D.
ln
-
.
sin(x)
-
.
+ C
-
.
ln
-
.
tan(x)
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.
+ C
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.
-ln
-
.
sin(x)
-
.
+ C
Solution
The integral of tan(x) is -ln|cos(x)| + C.
Correct Answer: A — -ln
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Q. What is the integral of tan(x)? (2021)
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A.
-ln
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B.
cos(x)
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C.
+ C
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D.
ln
-
.
sin(x)
-
.
+ C
-
.
ln
-
.
tan(x)
-
.
+ C
-
.
-ln
-
.
sin(x)
-
.
+ C
Solution
The integral of tan(x) is -ln|cos(x)| + C.
Correct Answer: A — -ln
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Q. What is the integral of x^2 with respect to x? (2021)
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A.
(1/3)x^3 + C
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B.
(1/2)x^2 + C
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C.
(1/4)x^4 + C
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D.
(1/5)x^5 + C
Solution
The integral of x^n is (1/(n+1))x^(n+1) + C. Here, n=2, so the integral is (1/3)x^3 + C.
Correct Answer: A — (1/3)x^3 + C
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