AP®MathsCollege BoardCalculus BCExam QuestionsUnit 6: Integration & Accumulation of ChangeMethods of Integration

Methods of Integration (College Board AP® Calculus BC): Exam Questions

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Methods of Integration

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1
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1 mark

integral 3 x squared sin open parentheses x cubed plus 4 close parentheses space d x equals

  • negative cos open parentheses x cubed plus 4 close parentheses plus C

  • cos open parentheses x cubed plus 4 close parentheses plus C

  • negative x cubed cos open parentheses x cubed plus 4 close parentheses plus C

  • x cubed cos open parentheses x cubed plus 4 close parentheses plus C

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2
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integral fraction numerator 4 x minus 3 over denominator 2 x squared minus 3 x plus 7 end fraction space d x equals

  • ln open vertical bar fraction numerator 1 over denominator 2 x squared minus 3 x plus 7 end fraction close vertical bar plus C

  • ln open vertical bar fraction numerator 4 x minus 3 over denominator 2 x squared minus 3 x plus 7 end fraction close vertical bar plus C

  • ln open vertical bar 2 x squared minus 3 x plus 7 close vertical bar plus C

  • fraction numerator 12 x squared minus 18 x over denominator 4 x cubed minus 9 x squared plus 42 x end fraction

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3
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integral subscript 0 superscript 1 open parentheses 4 x plus 6 close parentheses open parentheses x squared plus 3 x minus 1 close parentheses cubed space d x equals

  • 20

  • 30

  • 40

  • 50

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4
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integral subscript 1 superscript 3 fraction numerator x squared over denominator x cubed plus 1 end fraction space d x equals

  • 1 third ln 3

  • 1 third ln 14

  • ln 3

  • ln 14

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5
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integral open parentheses cos open parentheses 3 x close parentheses plus sin open parentheses 3 x close parentheses close parentheses d x equals

  • 1 third sin open parentheses 3 x close parentheses minus 1 third cos open parentheses 3 x close parentheses plus C

  • negative 1 third sin open parentheses 3 x close parentheses plus 1 third cos open parentheses 3 x close parentheses plus C

  • 3 sin open parentheses 3 x close parentheses minus 3 cos open parentheses 3 x close parentheses plus C

  • negative 3 sin open parentheses 3 x close parentheses plus 3 cos open parentheses 3 x close parentheses plus C

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6
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1 mark

integral subscript 0 superscript 2 e to the power of negative 3 x end exponent d x equals

  • negative e to the power of negative 6 end exponent over 3

  • negative 3 e to the power of negative 6 end exponent

  • e to the power of negative 6 end exponent minus 1

  • table row blank blank cell fraction numerator 1 minus e to the power of negative 6 end exponent over denominator 3 end fraction end cell end table

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7
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1 mark

Using the substitution u equals 3 x plus 2, integral subscript 2 superscript 8 open parentheses 3 x plus 2 close parentheses to the power of 7 d x is equivalent to

  • 1 third integral subscript 0 superscript 2 u to the power of 7 d u

  • 1 third integral subscript 8 superscript 26 u to the power of 7 d u

  • integral subscript 0 superscript 2 u to the power of 7 d u

  • integral subscript 8 superscript 26 u to the power of 7 d u

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8
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integral subscript 0 superscript 1 open parentheses 2 x minus 1 close parentheses to the power of 4 d x equals

  • 0

  • 1 fifth

  • 2 over 5

  • 4 over 5

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9
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1 mark

integral x e to the power of x space d x equals

  • e to the power of x plus C

  • 1 half x squared e to the power of x plus C

  • x e to the power of x plus e to the power of x plus C

  • x e to the power of x minus e to the power of x plus C

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10
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integral fraction numerator 4 over denominator open parentheses x minus 2 close parentheses open parentheses x plus 2 close parentheses end fraction space d x equals

  • ln vertical line x minus 2 vertical line plus ln vertical line x plus 2 vertical line plus C

  • ln vertical line x minus 2 vertical line minus ln vertical line x plus 2 vertical line plus C

  • ln vertical line x plus 2 vertical line minus ln vertical line x minus 2 vertical line plus C

  • 4 ln vertical line open parentheses x minus 2 close parentheses open parentheses x plus 2 close parentheses vertical line plus C

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11
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1 mark

Let R be the region between the graph of y equals e to the power of negative 3 x end exponent and the x-axis for x greater or equal than 2. The area of R is:

  • e to the power of 6 over 3

  • 3 over e to the power of 6

  • 1 over e to the power of 6

  • fraction numerator 1 over denominator 3 e to the power of 6 end fraction

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1
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1 mark

Using the substitution u equals square root of x, integral subscript 4 superscript 9 fraction numerator cos square root of x over denominator square root of x end fraction space d x is equal to which of the following?

  • 1 half integral subscript 2 superscript 3 cos u space d u

  • 2 integral subscript 2 superscript 3 cos u space d u

  • integral subscript 4 superscript 9 cos u space d u

  • 2 integral subscript 4 superscript 9 cos u space d u

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2
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integral fraction numerator x over denominator x minus 1 end fraction space d x equals

  • x minus ln open vertical bar x close vertical bar plus C

  • negative x minus ln open vertical bar x minus 1 close vertical bar plus C

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

  • fraction numerator x squared over denominator x squared minus 2 x end fraction plus c

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3
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integral subscript pi over 2 end subscript superscript pi fraction numerator sin x over denominator square root of 1 minus cos x end root end fraction d x equals

  • negative 2 open parentheses square root of 2 plus 1 close parentheses

  • negative 2 open parentheses square root of 2 minus 1 close parentheses

  • 2 open parentheses square root of 2 minus 1 close parentheses

  • 2 open parentheses square root of 2 plus 1 close parentheses

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4
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1 mark

integral fraction numerator 1 over denominator x squared minus 6 x plus 10 end fraction d x equals

  • arcsin open parentheses x minus 3 close parentheses plus straight C

  • arc tan open parentheses x minus 3 close parentheses plus straight C

  • ln open vertical bar x squared minus 6 x plus 10 close vertical bar plus C

  • 1 over open parentheses x squared minus 6 x plus 10 close parentheses squared plus C

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5
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integral fraction numerator x cubed plus 2 x squared plus 2 x plus 2 over denominator x plus 1 end fraction d x equals

  • fraction numerator 3 x cubed plus 8 x squared plus 12 x plus 24 over denominator 6 x plus 12 end fraction plus C

  • 1 third x cubed plus 1 half x squared plus 2 x plus C

  • table row blank blank cell 1 third x cubed plus 1 half x squared plus x plus ln open vertical bar x plus 1 close vertical bar plus C end cell end table

  • open parentheses 1 fourth x to the power of 4 plus 2 over 3 x cubed plus x squared plus 2 x close parentheses times ln open vertical bar x plus 1 close vertical bar plus C

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6
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integral fraction numerator x cubed over denominator x to the power of 4 minus 9 end fraction d x equals

  • fraction numerator negative 1 over denominator 8 open parentheses x to the power of 4 minus 9 close parentheses squared end fraction plus C

  • 1 fourth ln open vertical bar x to the power of 4 minus 9 close vertical bar plus C

  • 4 ln open vertical bar x to the power of 4 minus 9 close vertical bar plus C

  • 1 third arctan open parentheses x squared over 3 close parentheses plus C

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7
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integral x cubed sin open parentheses x to the power of 4 close parentheses d x equals

  • negative 1 fourth cos open parentheses x to the power of 4 close parentheses plus C

  • 1 fourth cos open parentheses x to the power of 4 close parentheses plus C

  • negative x cubed over 4 cos open parentheses x to the power of 4 close parentheses plus C

  • negative x to the power of 4 over 4 cos open parentheses x to the power of 5 over 5 close parentheses plus C

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8
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integral fraction numerator x squared over denominator square root of 5 x cubed plus 1 end root end fraction d x equals

  • table row blank blank cell 1 over 30 open parentheses 5 x cubed plus 1 close parentheses to the power of 1 half end exponent plus C end cell end table

  • table row blank blank cell 2 over 15 open parentheses 5 x cubed plus 1 close parentheses to the power of 1 half end exponent plus C end cell end table

  • table row blank blank cell 2 over 3 x cubed open parentheses 5 x cubed plus 1 close parentheses to the power of 1 half end exponent plus C end cell end table

  • table row blank blank cell 2 over 45 open parentheses 5 x cubed plus 1 close parentheses to the power of 3 over 2 end exponent plus C end cell end table

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9
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integral x cubed space ln open parentheses 3 x close parentheses space d x space equals

  • 1 fourth x to the power of 4 space open parentheses ln open parentheses 3 x close parentheses minus 3 close parentheses plus C

  • 1 over 16 x to the power of 4 space open parentheses 4 ln open parentheses 3 x close parentheses minus 3 close parentheses plus C

  • 1 over 16 x to the power of 4 space open parentheses 4 ln open parentheses 3 x close parentheses space minus 1 close parentheses plus C

  • 1 third x cubed space open parentheses 3 ln open parentheses 3 x close parentheses minus 1 close parentheses plus C

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10
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integral fraction numerator x minus 1 over denominator x squared minus 5 x plus 6 end fraction space d x equals

  • ln vertical line x minus 2 vertical line minus 2 ln vertical line x minus 3 vertical line plus C

  • 2 ln vertical line x minus 3 vertical line plus ln vertical line x minus 2 vertical line plus C

  • 2 ln vertical line x minus 3 vertical line minus ln vertical line x minus 2 vertical line plus C

  • 1 half ln vertical line x squared minus 5 x plus 6 vertical line plus C

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11
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1 mark

Evaluate the improper integral integral subscript negative infinity end subscript superscript infinity f open parentheses x close parentheses space italic d x where

f open parentheses x close parentheses open curly brackets table row cell e to the power of x comma space space space space space space x less than 0 end cell row cell e to the power of negative x end exponent comma space space space space x greater or equal than 0 end cell end table close

  • negative 2

  • 1

  • 2

  • infinity

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1
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integral fraction numerator x over denominator square root of 1 minus 25 x squared end root end fraction space d x equals

  • 1 fifth arcsin open parentheses 5 x close parentheses plus C

  • x over 5 arcsin open parentheses 5 x close parentheses plus C

  • negative 1 over 25 square root of 1 minus 25 x squared end root plus C

  • negative 1 over 25 ln square root of 1 minus 25 x squared end root plus C

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2
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1 mark

Let f be a function such that integral subscript 3 superscript 12 f open parentheses 3 x close parentheses d x equals 9. Which of the following must be true?

  • integral subscript 1 superscript 4 f open parentheses t close parentheses space d t equals 3

  • integral subscript 1 superscript 4 f open parentheses t close parentheses space d t equals 27

  • integral subscript 9 superscript 36 f open parentheses t close parentheses space d t equals 3

  • integral subscript 9 superscript 36 f open parentheses t close parentheses space d t equals 27

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3
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1 mark

If f is a continuous function and if F to the power of apostrophe open parentheses x close parentheses equals f open parentheses x close parentheses for all real numbers x, then integral subscript 4 superscript 9 6 x f open parentheses x squared close parentheses d x equals

  • table row blank blank cell fraction numerator F open parentheses 3 close parentheses over denominator 6 end fraction minus fraction numerator F open parentheses 2 close parentheses over denominator 6 end fraction end cell end table

  • table row blank blank cell 3 F open parentheses 3 close parentheses minus 3 F open parentheses 2 close parentheses end cell end table

  • table row blank blank cell fraction numerator F open parentheses 81 close parentheses over denominator 6 end fraction minus fraction numerator F open parentheses 16 close parentheses over denominator 6 end fraction end cell end table

  • table row blank blank cell 3 F open parentheses 81 close parentheses minus 3 F open parentheses 16 close parentheses end cell end table

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4
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1 mark

integral open parentheses square root of 1 minus cos x end root space integral subscript 0 superscript x cos t space d t close parentheses space d x equals

  • cos x square root of 1 minus cos x end root plus C

  • open parentheses cos x minus 1 close parentheses square root of 1 minus cos x end root plus C

  • 3 over 2 open parentheses 1 minus cos x close parentheses plus C

  • table row blank blank cell 2 over 3 open parentheses 1 minus cos x close parentheses to the power of 3 over 2 end exponent plus C end cell end table

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5
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1 mark

integral fraction numerator 1 over denominator square root of 3 plus 2 x minus x squared end root end fraction d x equals

  • arc sin open parentheses fraction numerator x minus 1 over denominator 2 end fraction close parentheses plus C

  • 2 arc sin open parentheses fraction numerator x minus 1 over denominator 2 end fraction close parentheses plus C

  • arc sin open parentheses x minus 1 close parentheses plus C

  • 2 arc sin open parentheses x minus 1 close parentheses plus C

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6
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1 mark

integral fraction numerator x to the power of 4 minus 2 x cubed plus x squared minus 7 over denominator x minus 2 end fraction d x equals

  • table row blank blank cell 1 fourth x to the power of 4 minus 2 over 3 x cubed plus 1 half x squared minus 7 ln open vertical bar x minus 2 close vertical bar plus C end cell end table

  • table row blank blank cell 1 fourth x to the power of 4 plus 1 half x squared plus 2 x minus 3 ln open vertical bar x minus 2 close vertical bar plus C end cell end table

  • table row blank blank cell x to the power of 4 plus 2 x squared plus 8 x minus 12 ln open vertical bar x minus 2 close vertical bar plus C end cell end table

  • table row blank blank cell 3 x to the power of 4 minus 8 x cubed plus 6 x squared minus 84 ln open vertical bar x minus 2 close vertical bar plus C end cell end table

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7
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integral x squared e to the power of 4 x end exponent space d x equals

  • table row blank blank cell 1 over 16 e to the power of 4 x end exponent open parentheses 4 x minus 1 close parentheses plus C end cell end table

  • 1 over 32 e to the power of 4 x end exponent open parentheses 8 x squared plus 4 x plus 1 close parentheses plus C

  • 1 over 32 e to the power of 4 x end exponent open parentheses 8 x squared minus 4 x plus 1 close parentheses plus C

  • 1 over 32 e to the power of 4 x end exponent open parentheses 8 x squared minus 4 x minus 1 close parentheses plus C

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8
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integral fraction numerator 2 x cubed minus 2 x squared minus 2 x minus 6 over denominator x squared minus x end fraction space d x equals

  • x squared plus 6 ln vertical line x vertical line minus 8 ln vertical line x minus 1 vertical line plus C

  • x squared minus 6 ln vertical line x vertical line plus 8 ln vertical line x minus 1 vertical line plus C

  • x squared plus 8 ln vertical line x vertical line minus 6 ln vertical line x minus 1 vertical line plus C

  • 6 ln vertical line x vertical line minus 8 ln vertical line x minus 1 vertical line plus C

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9
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1 mark

Evaluate the integral integral subscript 0 superscript 2 fraction numerator 1 over denominator square root of vertical line x minus 1 vertical line end root end fraction space italic d x.

  • 0

  • 2

  • 4

  • The integral is divergent

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