Integration - Quiz 3

Question 1

Which of the following is the indeﬁnite integral of $2 2x + 3?$

Not correct. Choice (a) is false.

Try again, you seem to be mixing up integration and differentiation.

Not correct. Choice (b) is false.

Try again, recall $∫ n -1--- n+1 x dx = n+ 1x ,$ and you should add a constant of integration.

Not correct. Choice (c) is false.

Try again, recall $∫ n --1-- n+1 x dx = n+ 1x .$

Your answer is correct.

Since $∫ xndx = --1-xn+1 n+ 1$ integrating term by term we have
$∫ 1 2 (2x2 + 4)dx = 2×-x3 + 3x+ c = -x3 + 3x +c , 3 3$ noting that we add a constant of integration.

Question 2

Which of the following is the indeﬁnite integral of $x14 ?$

Not correct. Choice (a) is false.

Try again, recall that $-1= 4. 54 5$

Your answer is correct.

Since $1+ 1 = 5 4 4$ we have $∫ 14 4 54 x dx = 5x +c .$

Not correct. Choice (c) is false.

Try again, recall that $1 5 4 + 1 = 4 .$

Not correct. Choice (d) is false.

Try again, you seem to have differentiated $x14 .$

Question 3

Which of the following is the indeﬁnite integral of $-3 x3 ?$

Not correct. Choice (a) is false.

Try again looking carefully at the sign.

Not correct. Choice (b) is false.

Try again, you have differentiated.

Your answer is correct.

$∫ ∫ 3-dx = 3x-3dx = - 3x-2 + c = --3 + c. x3 2 2x2$

Not correct. Choice (d) is false.

Try again, you have almost differentiated, watch the sign of the index.

Question 4

Which of the following correctly evaluates the deﬁnite integral $∫ 2 (x-2 +2x -3)dx? 1$

Not correct. Choice (a) is false.

Try again, you seem to have added the two components of the deﬁnite integral instead of subtracting them. Note also that the answer must be a positive number.

Not correct. Choice (b) is false.

Try again, you appear not to have added the fractions together correctly. Note that $a- c a+-c- b + d ⁄= b+ d .$

Your answer is correct.

$∫ 2 -2 -3 1 (x + 2x dx$	=	$[ -1 2 -2]2 - x - 2x 1$
	=	$1 1 3 5 [-2 - 4]- [- 1- 1] = - 4 + 2 = 4 .$

Not correct. Choice (d) is false.

Try again, you appear not to have integrated correctly. Note that $∫ ∫ 1 √-- 3√-- 1 12 13 0 ( x + x) dx = 0 (x + x )dx.$

Question 5

Which of the following correctly ﬁnds the area of the region bounded by the graph of $x2 + 2,$ the $x$ -axis and the lines $x = - 1$ and $x = 2?$

Your answer is correct.

The calculation we must perform is

$∫ 2 2 (x + 4)dx -1$	=	$[1 3 ]2 3x + 2x -1$
	=	$1 3 1 3 [3 × 2 + 2 × 2]- [3 × (- 1) + 2× (- 1)]$
	=	$8 + 4- (- 1 - 2) = 8 + 4+ 1 + 2 = 9 . 3 3 3 3$

Not correct. Choice (b) is false.

Try again, you may have differentiated instead of integrated.
The calculation we must perform is $∫ 2 2 (x + 2)dx. -1$

Not correct. Choice (c) is false.

Try again, you may have made a mistake with a sign. The calculation we must perform is $∫ 2 2 (x + 2)dx. -1$

Not correct. Choice (d) is false.

Try again, you may have made a mistake with a sign. The calculation we must perform is $∫ 2 2 -1(x + 2)dx.$


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