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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
Sme Calculator1 mark

Find the indefinite integral integral open parentheses 2 x to the power of 7 minus square root of x plus 1 over x squared close parentheses space d x.

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2
Sme Calculator2 marks

Given that f apostrophe open parentheses x close parentheses equals sin open parentheses 2 x close parentheses minus cos open parentheses 4 x close parentheses and that f open parentheses pi close parentheses equals 3 over 2, find f open parentheses x close parentheses.

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3
Sme Calculator2 marks

Given that r apostrophe open parentheses x close parentheses equals e to the power of x minus 2 to the power of x, and that r open parentheses 0 close parentheses equals negative fraction numerator 1 over denominator ln 2 end fraction, find an expression for r open parentheses x close parentheses.

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4
Sme Calculator2 marks

Find the indefinite integral integral open parentheses sec squared x plus fraction numerator 3 over denominator 1 plus x squared end fraction close parentheses space d x.

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5
Sme Calculator2 marks

Find the indefinite integral integral fraction numerator 3 x to the power of 4 plus 5 x squared minus 2 x to the power of 7 plus square root of x over denominator x to the power of 4 end fraction d x.

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1
Sme Calculator1 mark

Work out the indefinite integral integral fraction numerator x squared plus x minus 12 over denominator x squared minus 3 x end fraction space d x.

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2
Sme Calculator1 mark

Work out the indefinite integral integral open parentheses x to the power of 4 minus x plus 1 over x close parentheses squared space d x.

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3
Sme Calculator2 marks

The rate of change of a curve is given by fraction numerator d y over denominator d x end fraction equals 3 x squared minus sin x. Given that the curve passes through the point open parentheses pi comma space 1 close parentheses, find the equation of the curve in terms of x.

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4
Sme Calculator2 marks

The function A is used to model the surface area of a pond, in square meters, that is covered by a particular species of invasive pond weed. The rate of change function for A is A to the power of apostrophe, with the rate of change at time t given by A to the power of apostrophe open parentheses t close parentheses equals e to the power of negative t over 2 end exponent. Given that 1 square meter of the pond is covered by the weed at at time t equals 0 , find an expression for A open parentheses t close parentheses.

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5
Sme Calculator3 marks

Given that h satisfies the equation h open parentheses x close parentheses equals integral open parentheses e to the power of 3 x end exponent minus 3 to the power of x plus 1 close parentheses space d x and that when x equals 1 the value of h is 1 third e cubed minus fraction numerator 3 over denominator ln 3 end fraction, find an explicit expression for h in terms of x.

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1
Sme Calculator2 marks

A function f satisfies the equation f to the power of apostrophe open parentheses x close parentheses equals fraction numerator x squared plus x plus 1 over denominator x cubed plus x end fraction, where f to the power of apostrophe is the derivative of f. Given that f open parentheses 1 close parentheses equals 0, find an expression for f open parentheses x close parentheses in terms of x.

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2
Sme Calculator2 marks

Given that for a particular curve fraction numerator d y over denominator d x end fraction equals open parentheses e to the power of 2 x end exponent plus e to the power of negative 2 x end exponent close parentheses squared over e to the power of 2 x end exponent and that the curve passes through the origin, find the equation of the curve.

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3
Sme Calculator3 marks

Given that f apostrophe open parentheses x close parentheses equals cos space x space csc squared x and f open parentheses pi over 2 close parentheses equals 0 find f open parentheses x close parentheses.

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4
Sme Calculator3 marks

Work out the indefinite integral integral open parentheses fraction numerator 1 over denominator square root of 4 minus 4 x squared end root end fraction plus fraction numerator 2 over denominator 1 plus x squared end fraction close parentheses space d x.

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5
Sme Calculator3 marks

Given that f apostrophe open parentheses x close parentheses equals 5 cross times 3 to the power of open parentheses 2 x plus 1 close parentheses end exponent and that f open parentheses 0 close parentheses equals 7, find f open parentheses x close parentheses.

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