Q. Calculate the integral ∫ (x^2 + 2x + 1) dx.
-
A.
(1/3)x^3 + x^2 + x + C
-
B.
(1/3)x^3 + x^2 + C
-
C.
(1/3)x^3 + 2x^2 + C
-
D.
(1/3)x^3 + x^2 + x
Solution
The integral of x^2 is (1/3)x^3, the integral of 2x is x^2, and the integral of 1 is x. Thus, ∫ (x^2 + 2x + 1) dx = (1/3)x^3 + x^2 + x + C.
Correct Answer: A — (1/3)x^3 + x^2 + x + C
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Q. Calculate the integral ∫ (x^2 + 2x + 1)/(x + 1) dx.
-
A.
(1/3)x^3 + x^2 + C
-
B.
x^2 + 2x + C
-
C.
x^2 + x + C
-
D.
(1/3)x^3 + (1/2)x^2 + C
Solution
The integrand simplifies to x + 1. Therefore, ∫ (x + 1) dx = (1/2)x^2 + x + C.
Correct Answer: A — (1/3)x^3 + x^2 + C
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Q. Calculate the integral ∫ (x^3 - 4x) dx.
-
A.
(1/4)x^4 - 2x^2 + C
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B.
(1/4)x^4 - 2x^2
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C.
(1/4)x^4 - 4x^2 + C
-
D.
(1/4)x^4 - 2x^2 + 1
Solution
The integral of x^3 is (1/4)x^4 and the integral of -4x is -2x^2. Therefore, ∫ (x^3 - 4x) dx = (1/4)x^4 - 2x^2 + C.
Correct Answer: A — (1/4)x^4 - 2x^2 + C
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Q. Calculate the integral ∫ cos^2(x) dx.
-
A.
(1/2)x + (1/4)sin(2x) + C
-
B.
(1/2)x + C
-
C.
(1/2)x - (1/4)sin(2x) + C
-
D.
(1/2)x + (1/2)sin(2x) + C
Solution
Using the identity cos^2(x) = (1 + cos(2x))/2, we find that ∫ cos^2(x) dx = (1/2)x + (1/4)sin(2x) + C.
Correct Answer: A — (1/2)x + (1/4)sin(2x) + C
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Q. Evaluate the integral ∫ (1/x) dx.
-
A.
ln
-
B.
x
-
C.
+ C
-
D.
ln(x) + C
-
.
1/x + C
-
.
x + C
Solution
The integral of 1/x is ln|x|. Therefore, ∫ (1/x) dx = ln|x| + C.
Correct Answer: A — ln
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Q. Evaluate the integral ∫ (2x + 1)/(x^2 + x) dx.
-
A.
ln
-
B.
x^2 + x
-
C.
+ C
-
D.
ln
-
.
x
-
.
+ C
-
.
ln
-
.
x^2 + x
-
.
+ 1 + C
-
.
ln
-
.
x^2 + x
-
.
+ 1
Solution
Using partial fraction decomposition, we can integrate to find that ∫ (2x + 1)/(x^2 + x) dx = ln|x^2 + x| + C.
Correct Answer: A — ln
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Q. Evaluate the integral ∫ (3x^2 + 2x + 1) dx.
-
A.
x^3 + x^2 + x + C
-
B.
x^3 + x^2 + C
-
C.
x^3 + x^2 + x
-
D.
3x^3 + 2x^2 + x + C
Solution
The integral of 3x^2 is x^3, the integral of 2x is x^2, and the integral of 1 is x. Therefore, ∫ (3x^2 + 2x + 1) dx = x^3 + x^2 + x + C.
Correct Answer: A — x^3 + x^2 + x + C
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Q. Evaluate the integral ∫ (sec^2(x)) dx.
-
A.
tan(x) + C
-
B.
sec(x) + C
-
C.
sin(x) + C
-
D.
cos(x) + C
Solution
The integral of sec^2(x) is tan(x) + C.
Correct Answer: A — tan(x) + C
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Q. Evaluate the integral ∫ (x^2 + 2x + 1)/(x + 1) dx.
-
A.
(1/3)x^3 + x^2 + C
-
B.
x^2 + x + C
-
C.
(1/3)x^3 + (1/2)x^2 + C
-
D.
x^2 + 2x + C
Solution
By simplifying the integrand, we can integrate to find that ∫ (x^2 + 2x + 1)/(x + 1) dx = (1/3)x^3 + x^2 + C.
Correct Answer: A — (1/3)x^3 + x^2 + C
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Q. Evaluate the integral ∫ cos(3x) dx.
-
A.
(1/3)sin(3x) + C
-
B.
sin(3x) + C
-
C.
(1/3)cos(3x) + C
-
D.
-(1/3)sin(3x) + C
Solution
The integral of cos(kx) is (1/k)sin(kx). Here, k = 3, so ∫ cos(3x) dx = (1/3)sin(3x) + C.
Correct Answer: A — (1/3)sin(3x) + C
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Q. Evaluate the integral ∫ cos(5x) dx.
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A.
1/5 sin(5x) + C
-
B.
-1/5 sin(5x) + C
-
C.
5 sin(5x) + C
-
D.
sin(5x) + C
Solution
The integral of cos(kx) is (1/k)sin(kx). Here, k = 5, so ∫ cos(5x) dx = (1/5)sin(5x) + C.
Correct Answer: A — 1/5 sin(5x) + C
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Q. Evaluate the integral ∫ e^(3x) dx.
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A.
(1/3)e^(3x) + C
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B.
(1/3)e^(3x)
-
C.
3e^(3x) + C
-
D.
e^(3x) + C
Solution
The integral of e^(kx) is (1/k)e^(kx). Here, k = 3, so ∫ e^(3x) dx = (1/3)e^(3x) + C.
Correct Answer: A — (1/3)e^(3x) + C
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Q. Find the integral ∫ (1/x) dx.
-
A.
ln
-
B.
x
-
C.
+ C
-
D.
x + C
-
.
1/x + C
-
.
e^x + C
Solution
The integral of 1/x is ln|x| + C.
Correct Answer: A — ln
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Q. Find the integral ∫ (2x + 1)/(x^2 + x) dx.
-
A.
ln
-
B.
x^2 + x
-
C.
+ C
-
D.
ln
-
.
x
-
.
+ C
-
.
ln
-
.
x^2 + x
-
.
+ 1
-
.
ln
-
.
x
-
.
+ 1
Solution
Using partial fraction decomposition, we can integrate to find that ∫ (2x + 1)/(x^2 + x) dx = ln|x^2 + x| + C.
Correct Answer: A — ln
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Q. Find the integral ∫ (tan(x))^2 dx.
-
A.
tan(x) - x + C
-
B.
tan(x) + x + C
-
C.
tan(x) + x
-
D.
tan(x) - x
Solution
Using the identity tan^2(x) = sec^2(x) - 1, we find that ∫ (tan(x))^2 dx = tan(x) - x + C.
Correct Answer: A — tan(x) - x + C
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Q. Find the integral ∫ (x^2 - 1)/(x - 1) dx.
-
A.
(1/3)x^3 - x + C
-
B.
(1/3)x^3 - x - 1 + C
-
C.
(1/3)x^3 - x + 1
-
D.
(1/3)x^3 - x - 1
Solution
The integrand simplifies to x + 1. Therefore, ∫ (x + 1) dx = (1/2)x^2 + x + C.
Correct Answer: A — (1/3)x^3 - x + C
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Q. Find the integral ∫ sin(2x) dx.
-
A.
-cos(2x)/2 + C
-
B.
cos(2x)/2 + C
-
C.
-sin(2x)/2 + C
-
D.
sin(2x)/2 + C
Solution
The integral of sin(kx) is -1/k * cos(kx). Here, k = 2, so ∫ sin(2x) dx = -cos(2x)/2 + C.
Correct Answer: A — -cos(2x)/2 + C
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