DP IB Maths: AA SL

Topic Questions

5.4 Further Integration

1a
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1 mark

Find the indefinite integral

integral sin space x space space straight d x

1b
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3 marks

Find the exact value for

integral subscript 1 superscript 4 1 over x space straight d x

1c
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2 marks

Find the indefinite integral for

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

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2a
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2 marks

Integrate

integral cos space 2 x space space straight d x

2b
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4 marks

Find the definite integral

integral subscript 0 superscript 2 open parentheses 3 x minus 1 close parentheses cubed space straight d x

2c
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2 marks

Find an expression for y given that

fraction numerator d y over denominator d x end fraction equals e to the power of 5 x end exponent

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3
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7 marks

Using a suitable substitution, show that

integral subscript 1 superscript 2 fraction numerator x over denominator x plus 4 end fraction space straight d x equals 1 plus 4 ln space 5 over 6

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4
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6 marks

Given that

cos space 2 theta space identical to 2 cos to the power of 2 space end exponent theta space minus 1

use calculus to find the exact value of

integral subscript straight pi over 4 end subscript superscript straight pi over 2 end superscript cos squared theta space d theta

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5a
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2 marks
Given that  straight f open parentheses x close parentheses equals 2 x cubed plus 4 x comma  find  straight f apostrophe open parentheses x close parentheses.
5b
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4 marks
Hence, or otherwise, find


integral fraction numerator 3 x squared plus 2 over denominator 2 x cubed plus 4 x end fraction space straight d x

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6a
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2 marks

The diagram below shows a sketch of the curves with equations

space space y equals x squared minus 3 x plus 4 and y equals 4 minus x squared plus 2 x

q11-8-2-further-integration-medium-a-level-maths-pure-screenshot

Find the x-coordinates of the intersections of the two graphs.

6b
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2 marks

Show that the area of the shaded region labelled R is given by

integral subscript 0 superscript 5 over 2 end superscript open parentheses 5 x minus 2 x squared close parentheses space d x

6c
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2 marks

Use calculus to find the area of the shaded region labelled R.

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7a
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3 marks

The diagram below shows the graphs of the line space y equals 6 minus x spaceand  the curve space y equals x squared.

GztGkQjs_q10-8-1-integration-easy-a-level-maths-pure-screenshot

Work out the x-coordinates of the points labelled P, Q and R.

7b
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4 marks

Work out the area of the shaded region.

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8a
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2 marks

Consider the function h left parenthesis x right parenthesis such that

integral subscript 1 superscript 5 h left parenthesis x right parenthesis straight d x equals 2.

Find

integral subscript 5 superscript 1 h open parentheses x close parentheses space straight d x

8b
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3 marks

Find

integral subscript 1 superscript 5 fraction numerator h open parentheses x close parentheses plus 1 over denominator 2 end fraction d x

8c
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3 marks

Find

integral subscript 1 superscript 5 open parentheses h open parentheses x close parentheses plus 2 x close parentheses straight d x

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9a
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3 marks

Consider the function f left parenthesis x right parenthesis equals ln left parenthesis 2 x squared plus 1 right parenthesis .

Find f to the power of apostrophe left parenthesis x right parenthesis.

9b
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3 marks

Hence, find

integral fraction numerator x over denominator 2 x squared plus 1 end fraction straight d x

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10
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5 marks

Let  f to the power of apostrophe left parenthesis x right parenthesis equals x to the power of 2 space end exponent cos space left parenthesis x cubed plus 1 right parenthesis.

Find f left parenthesis x right parenthesis space spacegiven that  f left parenthesis negative 1 right parenthesis equals 1.

 

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