AP®MathsCollege BoardCalculus ABExam QuestionsUnit 6: Integration & Accumulation of ChangeIntegration & Antiderivatives

Integration & Antiderivatives (College Board AP® Calculus AB): Exam Questions

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Integration & Antiderivatives

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1
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1 mark

integral csc x space cot x space d x equals

  • negative csc x plus C

  • negative cot x plus C

  • negative fraction numerator csc squared x over denominator 2 end fraction plus C

  • negative fraction numerator cot squared x over denominator 2 end fraction plus C

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2
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integral sec squared x space d x equals

  • 1 third sec cubed x plus C

  • csc squared x plus C

  • sin squared x plus C

  • tan x plus C

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3
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1 mark

integral 1 over x space d x equals

  • C

  • x plus C

  • 1 half x squared plus C

  • ln open vertical bar x close vertical bar plus C

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4
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1 mark

integral sin x space d x equals

  • cos x plus C

  • negative cos x plus C

  • 1 half sin squared x plus C

  • negative 1 half sin squared x plus C

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5
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1 mark

integral fraction numerator 1 over denominator 1 plus x squared end fraction space d x equals

  • open parentheses 1 plus x squared close parentheses to the power of negative 1 end exponent plus C

  • open parentheses 1 plus x squared close parentheses to the power of negative 2 end exponent plus C

  • arcsin x plus C

  • arc tan x plus C

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6
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1 mark

integral 1 over x cubed d x equals

  • ln open parentheses x cubed close parentheses plus C

  • negative 1 half x to the power of negative 2 end exponent plus C

  • x to the power of negative 3 end exponent plus C

  • negative 3 x to the power of negative 4 end exponent plus C

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1
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1 mark

integral fraction numerator 6 x squared minus 5 over denominator 2 x end fraction space d x equals

  • 3 x minus fraction numerator 5 over denominator 2 x end fraction plus C

  • fraction numerator 2 x cubed minus 5 x over denominator x squared end fraction plus C

  • 3 over 2 x squared minus 5 e to the power of 2 x end exponent plus C

  • 3 over 2 x squared minus 5 over 2 ln open vertical bar x close vertical bar plus C

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2
1 mark

integral open parentheses 3 x squared plus 1 close parentheses open parentheses 2 x minus 1 close parentheses space d x equals

  • 18 x squared minus 6 x plus 2 plus C

  • 6 x cubed minus 3 x squared plus 2 x minus 1 plus C

  • 3 over 2 x to the power of 4 minus x cubed plus x squared minus x plus C

  • x to the power of 5 minus x to the power of 4 plus x cubed minus x squared plus C

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3
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integral fraction numerator 1 over denominator square root of x end fraction open parentheses fraction numerator x plus 1 over denominator x end fraction close parentheses space d x equals

  • negative fraction numerator 2 over denominator 3 x square root of x end fraction minus fraction numerator 2 over denominator 5 x squared square root of x end fraction plus C

  • 2 square root of x minus fraction numerator 2 over denominator square root of x end fraction plus C

  • 2 square root of x plus fraction numerator 4 over denominator square root of x end fraction plus C

  • 2 x square root of x plus 2 square root of x ln x plus C

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4
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1 mark

The curve with equation y equals f open parentheses x close parentheses goes through the point open parentheses 3 comma space 5 close parentheses. Given that fraction numerator d y over denominator d x end fraction equals 2 x squared minus 1, what is f open parentheses x close parentheses?

  • 2 over 3 x cubed minus x plus 5

  • 2 over 3 x cubed minus x minus 10

  • x squared minus x minus 1

  • 4 x minus 7

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5
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1 mark

The derivative of a function f is f to the power of apostrophe open parentheses x close parentheses equals fraction numerator 3 over denominator 2 square root of x end fraction minus 2 x.

Given that f open parentheses 4 close parentheses equals negative 8, what is f open parentheses x close parentheses?

  • f open parentheses x close parentheses equals x to the power of 3 over 2 end exponent minus x squared

  • f open parentheses x close parentheses equals 3 square root of x minus x squared plus 2

  • f open parentheses x close parentheses equals 3 square root of x minus x squared

  • f open parentheses x close parentheses equals 3 square root of x minus x squared minus 8

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6
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1 mark

A curve has slope 4 x minus 5 at each point open parentheses x comma space y close parentheses on the curve. Given that this curve passes through the point open parentheses 1 comma space 3 close parentheses, which of the following is an equation for the curve.

  • y equals negative x plus 4

  • y equals 2 x squared plus 1

  • y equals 2 x squared minus 5 x plus 3

  • y equals 2 x squared minus 5 x plus 6

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1
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1 mark

If f is a linear function, then integral f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses space d x equals

  • 0

  • C

  • f to the power of apostrophe open parentheses x close parentheses

  • f open parentheses x close parentheses

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2
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1 mark

integral fraction numerator x squared over denominator x squared plus 1 end fraction space d x equals

  • x minus arctan x plus C

  • ln open vertical bar x squared plus 1 close vertical bar plus C

  • fraction numerator 2 x over denominator open parentheses x squared plus 1 close parentheses squared end fraction

  • fraction numerator x cubed over denominator x cubed plus 3 x end fraction

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3
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1 mark

A function f has its derivative given by f to the power of apostrophe open parentheses x close parentheses equals fraction numerator 3 minus x over denominator x squared end fraction. Given that f open parentheses 1 close parentheses equals 2, what is f open parentheses x close parentheses?

  • table row blank blank cell negative 3 over x minus ln open vertical bar x close vertical bar end cell end table

  • table row blank blank cell 5 minus 3 over x minus ln open vertical bar x close vertical bar end cell end table

  • fraction numerator 18 minus 3 x over denominator 2 x squared end fraction

  • fraction numerator 18 minus 3 x minus 11 x squared over denominator 2 x squared end fraction

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4
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1 mark

A function f satisfies f open parentheses x close parentheses equals integral fraction numerator 2 plus 5 square root of 1 minus x squared end root over denominator square root of 1 minus x squared end root end fraction d x. Given that f open parentheses 1 close parentheses equals 5, what is the value of f open parentheses 0 close parentheses?

  • negative 5

  • negative pi

  • negative pi over 2

  • 1

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5
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1 mark

The curve with equation y equals f open parentheses x close parentheses goes through the point open parentheses 0 comma space minus fraction numerator 45 over denominator 32 ln 2 end fraction close parentheses. Given that fraction numerator d y over denominator d x end fraction equals 3 times 2 to the power of 4 x minus 3 end exponent, what is the value of f open parentheses 1 close parentheses?

  • negative fraction numerator 3 over denominator 2 ln 2 end fraction

  • 0

  • 1

  • fraction numerator 45 over denominator 32 ln 2 end fraction

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