Integration - Quiz 2

Question 1

Which of the following is the indeﬁnite integral of $2x3 + 4?$

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 $∫ xndx = --1--xn+1, n + 1$ and you should add a constant of integration.

Your answer is correct.

Since $∫ 1 xn dx = ----xn+1 n+ 1$ integrating term by term we have
$∫ 1 1 (2x3 + 4)dx = 2×-x4 + 4x+ c = -x4 + 4x +c , 4 2$ noting that we add a constant of integration.

Not correct. Choice (d) is false.

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

Question 2

Which of the following is the indeﬁnite integral of $1 2x 2 ?$

Your answer is correct.

Since $1 3 2 + 1 = 2$ we have $∫ 1 2 3 4 3 2x2 dx = 2× 3 x2 + c = 3x2 + c.$

Not correct. Choice (b) is false.

Try again, recall that $1 2 3-= 3 . 2$

Not correct. Choice (c) is false.

Try again, recall that $1 3 2 + 1 = 2 .$

Not correct. Choice (d) is false.

Try again, you seem to have differentiated $2x 12 .$

Question 3

Which of the following is the indeﬁnite integral of $-3 x4 ?$

Not correct. Choice (a) is false.

Try again looking carefully at the sign.

Not correct. Choice (b) is false.

Try again, you have differentiated.

Not correct. Choice (c) is false.

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

Your answer is correct.

$∫ ∫ 3-dx = 3x-4dx = - 3x-3 +c = --1 + c. x4 3 x3$

Question 4

Which of the following correctly evaluates the deﬁnite integral $∫ 1√ -- √3-- 0 ( x+ x)dx?$

Not correct. Choice (a) is false.

Try again, you seem to have differentiated the components rather than integrated them.

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.

$∫ 1 (√x-+ 3√x) dx 0$	=	$∫ 1 (x12 + x 13) dx 0$
	=	$[ ] 2 32 3 43 1 3x + 4 x 0$
	=	$[2+ 3]- [0] =-8 + -9 = 17. 3 4 12 12 12$

Not correct. Choice (d) is false.

Try again, you appear not to have integrated correctly.
Note that $∫ 1√ -- √-- ∫ 1 ( x + 3x)dx = (x 12 + x13)dx . 0 0$

Question 5

Which of the following correctly ﬁnds the area of the region bounded by the graph of $x2 - 4x +4 ,$ the $x$ -axis and the lines $x = 1$ and $x = 4?$

Not correct. Choice (a) is false.

Try again, you may have differentiated instead of integrated.
The calculation we must perform is $∫ 4 (x2 - 4x + 4)dx. 1$

Your answer is correct.

The calculation we must perform is

$∫ 4 (x2 - 4x + 4) dx 1$	=	$[ ]4 1x3 - 2x2 + 4x 3 1$
	=	$[1× 43 - 2 × 42 + 4× 4× 4]- [1 - 2 + 4] 3 3$
	=	$64 1 -3 - 32+ 16- (3 + 2) = 21- 16- 2 = 3.$

Not correct. Choice (c) is false.

Try again, you may have made a mistake with a sign. The calculation we must perform is $∫ 4 (x2 - 4x+ 4)dx . 1$

Not correct. Choice (d) is false.

Try again, you may have made a mistake with a sign. The calculation we must perform is $∫ 4 (x2 - 4x+ 4)dx. 1$


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