MTH108 CMA01 Answers

 

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2020_JUL_T2_PT_6

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CMA01

• Due 5 Oct at 23:55
 
• Points 100
 
• Questions 20
 
• Available 28 Sep at 0:00 - 5 Oct at 23:55 8 days
 
• Time limit None

Instructions

This Computer-Marked Assignment (CMA) covers topics from the first 3 lectures of this MTH108 course, with a heavy emphasis on techniques of integration.

There are 20 MCQ questions, each worth 5 marks.

Attempt history

	Attempt	Time	Score
LATEST	Attempt 1	158 minutes	95 out of 100

 Correct answers are hidden.

Score for this quiz: 95 out of 100

Submitted 30 Sep at 21:29

This attempt took 158 minutes.

 

Question 1

5 / 5 pts

Let  $f$ ( x ) = ln ⁡ ( 1 + x 2 ) for  $x$ ≤ 0 .

Determine the rule for $f$ − 1 ( x ) .

  

$f$ − 1 ( x ) = − e x − 1

Well done

  

$f$ − 1 ( x ) = 2 x 1 + x 2

  

$f$ − 1 ( x ) = 1 ln ⁡ ( 1 + x 2 )

  

$f$ − 1 ( x ) = e x − 1

 

Question 2

5 / 5 pts

Let $F$ ( x ) = arcsin ⁡ x for $0$ < x < 1. Compute $F$ ′ ( x ) for $0$ < x < 1.

  

$1$ x − x 2

  

$1$ 2 x − x 2$1$ 2 x − x 2$1$ 2 x − x 2

Well done

  

$arccos$ ⁡ x 2 x

  

$1$ 1 − x$1$ 1 − x$1$ 1 − x

 

Question 3

5 / 5 pts

Compute the indefinite integral 

$\int$ ( − 6 x 2 − 5 cos 2 ⁡ x ) d x

  

$6$ x + 5 cos ⁡ x + C$6$ x + 5 cos ⁡ x + C

  

$6$ x − 5 tan ⁡ x + C$6$ x − 5 tan ⁡ x + C

Well done

  

$6$ x + 5 tan ⁡ x + C$6$ x + 5 tan ⁡ x + C

  

$-$ 6 x − 5 tan ⁡ x + C$-$ 6 x − 5 tan ⁡ x + C

 

Question 4

5 / 5 pts

Compute the indefinite integral $\int$ ( x − 1 x 2 + 1 ) d x

  

$ln$ ⁡ ( 1 + x 2 ) 2 − arctan ⁡ x + C$ln$ ⁡ ( 1 + x 2 ) 2 − arctan ⁡ x + C$ln$ ⁡ ( 1 + x 2 ) 2 − arctan ⁡ x + C$ln$ ⁡ ( 1 + x 2 ) 2 − arctan ⁡ x + C

Well done

  

$ln$ ⁡ ( 1 + x 2 ) − arctan ⁡ x + C

  

$arctan$ ( 1 + x 2 ) $+$ C

  

$arctan$ ⁡ ( 1 + x 2 ) 2 + C

 

Question 5

5 / 5 pts

Compute the indefinite integral $\int$ sin 3 ⁡ x d x

  

$cos$ ⁡ 3 x 3 − cos ⁡ x + C$cos$ ⁡ 3 x 3 − cos ⁡ x + C

  

$sin$ ⁡ 4 x 4 − sin ⁡ 4 x + C$sin$ ⁡ 4 x 4 − sin ⁡ 4 x + C

  

$cos$ 3 ⁡ x 3 − cos ⁡ x + C$cos$ 3 ⁡ x 3 − cos ⁡ x + C

Well done

  

$sin$ 4 ⁡ x 4 + C$sin$ 4 ⁡ x 4 + C

 

Question 6

5 / 5 pts

In the computation of the indefinite integral

$\int$ ( ln ⁡ x ) 2 d x

which of the following techniques is used?

  

Integration by Partial Fractions

 

  

Product Rule

 

  

Integration by Substitution

 

  

Integration by Parts

 

Well done

 

Question 7

5 / 5 pts

In the computation of the indefinite integral

$\int$ 2 x 3 − 3 x 2 + 4 x x 2 − 2 x + 2 d x

which of the following techniques is used?

  

Quotient Rule

  

Long Division of Polynomials

Well done.

After Long Division is done, how would you proceed to get the final answer?

  

Integration by Parts

  

Integration by Partial Fractions

 

Question 8

5 / 5 pts

In the solution of the integral   

$\int$ x 1 − x 4 d x   

which of the following substitutions is used?

  

$u$ = x

  

$u$ = 1 − x 4

  

$u$ = x 4

  

$u$ = x 2

Well done!

 

Question 9

5 / 5 pts

In the computation of the indefinite integral

$\int$ sin 2 ⁡ ( 3 x ) d x

which of the following trigonometric identities is used?

  

$sin$ 2 ⁡ θ + cos 2 ⁡ θ = 1$sin$ 2 ⁡ θ + cos 2 ⁡ θ = 1

  

$cos$ ⁡ 2 θ = cos 2 ⁡ θ − sin 2 ⁡ θ$cos$ ⁡ 2 θ = cos 2 ⁡ θ − sin 2 ⁡ θ

  

$cos$ ⁡ 2 θ = 1 − 2 sin 2 ⁡ θ$cos$ ⁡ 2 θ = 1 − 2 sin 2 ⁡ θ

Well done.

In particular, we use this identity to achieve

$sin$ 2 ⁡ ( 3 x ) = 1 − cos ⁡ 6 x 2$sin$ 2 ⁡ ( 3 x ) = 1 − cos ⁡ 6 x 2

where the right hand side can be integrated easily.

  

$cos$ ⁡ 2 θ = 1 − sin 2 ⁡ θ$cos$ ⁡ 2 θ = 1 − sin 2 ⁡ θ

 

Question 10

5 / 5 pts

If $8$ x + 5 3 x 2 − 5 x − 2 = A 3 x + 1 + B x − 2, then

  

$A=1,\; B=3$$A=1,\; B=3$

  

$A$ = − 1 , B = 3$A$ = − 1 , B = 3

Well done

  

$A=2,\; B=3$$A=2,\; B=3$

  

$A=2,\; B=0$$A=2,\; B=0$

 

Question 11

5 / 5 pts

Evaluate the definite integral $\int$ 0 π ( 1 + tan ⁡ x cos ⁡ x ) d x

  

$-$ π$-$ π$-$ π$-$ π

  

$\pi$$\pi$$\pi$$\pi$

  

$\pi$ − 2

  

$\pi$ + 2$\pi$ + 2$\pi$ + 2

Well done

 

Question 12

5 / 5 pts

Evaluate the definite integral $\int$ 0 6 1 x 2 + 36 d x.

 

  

$1$$1$$1$

  

$\pi$ 2$\pi$ 2$\pi$ 2

  

$\pi$ 6

  

$\pi$ 24$\pi$ 24$\pi$ 24

Well done

 

Question 13

5 / 5 pts

Evaluate the definite integral $\int$ 0 1 x e − 2 x d x

  

$1$ − e − 2 3$1$ − e − 2 3$1$ − e − 2 3$1$ − e − 2 3

  

$1$ − 3 e − 2 4$1$ − 3 e − 2 4$1$ − 3 e − 2 4$1$ − 3 e − 2 4

Well done

  

$1$ − 3 e − 2 3

  

$1$ − e − 2 4

 

Question 14

5 / 5 pts

Compute the definite integral $\int$ 0 1 cos ⁡ π x d x

  

$1$ π$1$ π$1$ π

  

$2$ π$2$ π$2$ π

  

0

 

Well done

  

$-$ 1 π$-$ 1 π$-$ 1 π

 

Question 15

5 / 5 pts

Compute the definite integral $\int$ 0 1 | cos ⁡ π x | d x.

  

$\pi$$\pi$$\pi$$\pi$

  

$2$ π$2$ π$2$ π$2$ π

Well done

  

$1$ π$1$ π$1$ π$1$ π

  

0

 

 

Question 16

5 / 5 pts

Evaluate the definite integral $\int$ 0 π 3 tan 3 ⁡ x d x.

 

  

$\pi$ 6$\pi$ 6

  

$3$ 2 − ln ⁡ 2$3$ 2 − ln ⁡ 2$3$ 2 − ln ⁡ 2$3$ 2 − ln ⁡ 2$3$ 2 − ln ⁡ 2$3$ 2 − ln ⁡ 2

Well done

  

$1$ − ln ⁡ 2$1$ − ln ⁡ 2$1$ − ln ⁡ 2$1$ − ln ⁡ 2$1$ − ln ⁡ 2$1$ − ln ⁡ 2

  

$ln$ ⁡ 2 2 − 3 4

 

Question 17

5 / 5 pts

Evaluate the definite integral $\int$ 0 1 arctan ⁡ x d x.

  

$\pi$ 4 − ln ⁡ 2 2$\pi$ 4 − ln ⁡ 2 2$\pi$ 4 − ln ⁡ 2 2$\pi$ 4 − ln ⁡ 2 2$\pi$ 4 − ln ⁡ 2 2

Well done

  

$\pi$ 4 + ln ⁡ 2 2$\pi$ 4 + ln ⁡ 2 2$\pi$ 4 + ln ⁡ 2 2$\pi$ 4 + ln ⁡ 2 2$\pi$ 4 + ln ⁡ 2 2

  

$ln$ ⁡ 2 2 − 1 4

  

$\pi$ 4 − 1 2

 

Question 18

5 / 5 pts

Let f be a continuous function on [-1, 1]. Define F on [-1, 1] by 

$F$ ( x ) = ∫ 0 x f ( t ) d t.

Compute F(0)

  

0

 

Well done

  

Does not exist

 

  

2

 

  

1

 

 

IncorrectQuestion 19

0 / 5 pts

Let f be a continuous function on [-1, 1]. Define F on [-1, 1] by 

$F$ ( x ) = ∫ 0 x f ( t ) d t. If $F$ ′ ( x ) = e x + 2 x + 1 for x in (-1, 1)

Evaluate f(0).

  

1

 

  

2

 

  

0

 

  

3

 

 

Question 20

5 / 5 pts

Let f be a function on $($ − ∞ , ∞ ) such that $\int$ f ( x ) d x = x 2 e 2 x + C. Compute f(x).

  

$x$ 2 e 2 x ( x + 1 )

  

$2$ e 2 x ( x + 1 )$2$ e 2 x ( x + 1 )$2$ e 2 x ( x + 1 )

  

$x$ e 2 x ( x + 1 )

  

$2$ x e 2 x ( x + 1 )$2$ x e 2 x ( x + 1 )$2$ x e 2 x ( x + 1 )

Well done

Quiz score: 95 out of 100

Submission details:

Time:	158 minutes
Current score:	95 out of 100
Kept score:	95 out of 100