Further Integration (DP IB Maths: AI HL)

Exam Questions

4 hours34 questions
1a
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1 mark

Find the indefinite integral for

integral sin space x space space d x

1b
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3 marks

Show that the exact value of the definite integral

integral subscript 1 superscript 4 1 over x d x

is  2 space ln invisible function application space 2.

1c
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2 marks

Find the indefinite integral for

integral 7 e to the power of 7 x d x

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2a
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2 marks

Integrate

integral cos invisible function application space 2 x space space d x

2b
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3 marks

Show that

integral left parenthesis 3 x minus 1 right parenthesis cubed d x equals 1 over 12 left parenthesis 3 x minus 1 right parenthesis to the power of 4 plus c

where c is a constant of integration.

2c
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3 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 to the power of x

and that y equals 1   when x equals 0 .

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3a
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3 marks

Find the indefinite integral for

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

3b
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3 marks

Find the indefinite integral for

integral space fraction numerator x to the power of 2 over 3 end exponent plus x to the power of 11 over 6 end exponent over denominator x squared end fraction d x 

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4a
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2 marks

Given that , f left parenthesis x right parenthesis equals 2 x cubed plus 4 x find f to the power of apostrophe left parenthesis x right parenthesis .

4b
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4 marks

Hence, or otherwise, find

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

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5a
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3 marks

Consider the function f left parenthesis x right parenthesis equals l n invisible function application space left parenthesis 2 x squared plus 1 right parenthesis .

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

5b
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3 marks

Hence, find

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

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6
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5 marks

Let f to the power of apostrophe left parenthesis x right parenthesis equals x squared cos space invisible function application left parenthesis x cubed plus 1 right parenthesis.

Find  f left parenthesis x right parenthesis   given that f left parenthesis negative 1 right parenthesis equals 1 .

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7a
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2 marks

Show that

fraction numerator tan space invisible function application x over denominator sin space x space cos space x end fraction equals fraction numerator 1 over denominator cos squared space space invisible function application x end fraction

7b
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3 marks

q7b_5-4-further-integration-medium-ib-ai-hl

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8a
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3 marks

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

q7-5-4-further-integration-medium-ib-aa-sl

Point P is the point of intersection of the curve  y equals x squared  with the ­x ­-axis.  Point Q  is the point of intersection of the curve y equals x squared with the line y equals 6 minus x spacefor which space x greater than 0.  Point R is the point of intersection of the line  y equals 6 minus x  with the x-axis.

Write down the x-coordinates of points P comma Q space and space R.

8b
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2 marks

Calculate the area of the shaded region.

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9a
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4 marks

The diagram below shows the graph of the function f which is defined by

f left parenthesis x right parenthesis equals x space sin space x comma space space space space space 0 less or equal than x less or equal than 2 pi

q9-5-4-further-integration-hard-ib-ai-hl

The shaded region in the diagram is the region enclosed by the x-axis and the graph of y equals f left parenthesis x right parenthesis.

Find the area of the part of the shaded region that lies above the x-axis.

9b
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3 marks

Find the area of the entire shaded region.

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10a
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6 marks

The diagram below depicts the design for a new company logo. The logo is formed by a circle centred on the origin, which is divided into two regions by the curve y equals f left parenthesis x right parenthesis where f is the function defined by space f left parenthesis x right parenthesis equals cos space fraction numerator 3 x over denominator 2   end fraction comma space space minus pi less or equal than x less or equal than pi. The points where the circle and the curve intersect lie on the x-axis, as shown.

q10-5-4-further-integration-hard-ib-ai-hl

The shaded region in the diagram is the region inside that circle that lies below the curve y equals f left parenthesis x right parenthesis. .

(i)
Write down the radius of the circle that forms the outer border of the logo.
 
(ii)
Hence determine the exact area of the shaded region.
10b
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2 marks

Find the percentage of the circular logo that is shaded.

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11a
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3 marks

The following diagram shows a part of the graph of the curve y space equals 1 fourth open parentheses x minus 2 close parentheses squared .  The shaded region is the region enclosed by the graph and the positive x- and y-axes.

 q10a_advanced-integration_medium_ib-maths-aa-hl

 

(i)
Find the coordinates of the points where the graph intersects the coordinate axes.

(ii)
For the part of the curve that forms the boundary of the shaded region, show that  x equals 2 minus 2 square root of y .
11b
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6 marks

Find the area of the shaded region

 
(i)
by calculating it as an area between the curve and the x-axis.

(ii)
by calculating it as an area between the curve and the y-axis.
11c
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5 marks

Find the volume of the solid formed when the shaded region is rotated 2 straight pi radians about the x-axis.

11d
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5 marks

Find the volume of the solid formed when the shaded region is rotated 2 straight pi radians about the y-axis.

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12a
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3 marks

The diagram below shows the cross-section of a bowl that a company is planning to begin producing.

 q11a_advanced-integration_medium_ib-maths-aa-hl

 

As indicated on the diagram, one of the sides of the bowl in the cross-section may be described by the curve y equals square root of x squared minus 36 end root,  where units for x and y are centimetres.  The cross-section is entirely symmetrical about the y-axis.  The flat circular bottom of the bowl has a diameter of 12 cm, and the vertical depth of the bowl is 6 cm.  For purposes of answering this question, the thickness of the bottom and sides of the bowl may be regarded as negligible.

Find the exact coordinates of the point marked straight A on the diagram.

12b
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4 marks

Show that the capacity of the bowl in cm3 is given by

 
straight pi integral subscript 0 superscript b open parentheses y squared plus 36 close parentheses straight d y
 

where b is a constant to be determined.

12c
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2 marks

Hence find the capacity of the bowl.

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1a
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2 marks

Consider the function  f defined by f open parentheses x close parentheses equals open parentheses x squared minus x minus 2 close parentheses open parentheses x minus 5 close parentheses comma space minus 2 less or equal than x less or equal than 4.

Find the coordinates of the points where the graph of  y equals f open parentheses x close parentheses intercepts the x-axis.

1b
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3 marks

Find the indefinite integral

integral open parentheses x squared minus x minus 2 close parentheses open parentheses x minus 5 close parentheses d x
1c
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2 marks

Use your answer to part (b) to calculate the area of the region enclosed by the graph of  y equals f open parentheses x close parentheses and the x-axis.

 

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2a
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2 marks

Find the indefinite integral for

integral cos open parentheses x over 2 close parentheses d x

 

2b
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2 marks

Find the indefinite integral for

integral 5 e to the power of 3 x end exponent 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 sin open parentheses x minus straight pi over 3 close parentheses

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3a
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2 marks

Find the indefinite integral

integral fraction numerator 3 over denominator 2 x end fraction d x
3b
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3 marks

Find an expression for given that

 fraction numerator d y over denominator d x end fraction equals e to the power of 2 x plus 3 end exponent plus 2

 and also that y equals 5 when x equals negative 3 over 2 .

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4a
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3 marks

Find the indefinite integral for

integral open parentheses 1 fourth square root of x minus fraction numerator 2 over denominator cube root of x end fraction close parentheses d x
4b
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3 marks

Find the indefinite integral for

integral fraction numerator x 5 over 6 minus 5 x 7 over 3 over denominator x square root of x end fraction d x

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5a
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6 marks

Consider the function f open parentheses x close parentheses equals ln open parentheses 3 x squared minus 12 x plus 1 close parentheses.

(i)
Find  f apostrophe open parentheses x close parentheses.

(ii)
Hence, find 
integral fraction numerator 16 minus 8 x over denominator 3 x squared minus 12 x plus 1 end fraction d x
5b
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5 marks

Let g apostrophe open parentheses x close parentheses equals open parentheses x squared minus 5 x plus 6 close parentheses sin open parentheses 2 x cubed minus 15 x squared plus 36 x minus straight pi over 3 close parentheses

Find g open parentheses x close parentheses given that  g open parentheses 0 close parentheses equals 1.

 

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6a
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2 marks

Work out the following indefinite integrals:

integral fraction numerator 2 over denominator 3 space C o s squared space 3 x end fraction d x

6b
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3 marks

integral fraction numerator 7 tan open parentheses x over 2 close parentheses over denominator 6 sin x over 2 cos x over 2 end fraction d x

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7
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6 marks

Use definite integration to find the exact value of

 integral subscript 2 superscript 5 fraction numerator x plus 1 over denominator x squared plus 2 x minus 5 end fraction d x 

giving your final answer in as simple a form as possible.

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8a
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2 marks

The diagram below shows the graph of the function f which is defined by

f open parentheses x close parentheses equals negative x cos space x comma space space space space space space space 0 less or equal than x less or equal than fraction numerator 3 straight pi over denominator 2 end fraction

mi-q8a-5-4-ib-ai-hl-further-integration-hard_dig

The shaded region in the diagram is the region enclosed by the x-axis and the graph of y equals f open parentheses x close parentheses

Explain why the area of the shaded region is not equal to

integral subscript 0 superscript fraction numerator 3 straight pi over denominator 2 end fraction end superscript minus x cos space x d x
8b
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3 marks

Find the area of the entire shaded region.

8c
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4 marks

Find the individual areas of the parts of the shaded region (i) above and (ii) below the x-axis.

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9a
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4 marks

The shaded region in the diagram below depicts the design for a new company logo.  The upper border of the logo is formed by a part of the curve with equation y equals fraction numerator 9 straight pi squared minus 16 x squared over denominator 32 end fraction,  while the lower border of the logo is formed by a part of the curve with equation  y equals negative 1 plus 2 space sin squared x.  The points where the two curves intersect lie on the x-axis, as shown.

mi-q9a-5-4-ib-ai-hl-further-integration-hard_dig

Find the exact coordinates of 

(i)
the points of intersection of the two curves 

(ii)
the other points where the lower border of the logo intersects the x-axis.
9b
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5 marks

Hence find the area of the company logo.

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10a
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3 marks

The following diagram shows a part of the graph of the curve  y equals k x minus x squared,  where k greater than 0  is a constant.  The point marked A is the vertex of the curve.  Region R is the region enclosed by the curve and the x-axis.  Region S is the region enclosed by the curve, the positive y-axis, and the line through point A with gradient zero.

 mi-q10a-5-4-ib-ai-hl-further-integration-hard_dig

Show that the part of the curve bordering the region  can also be represented by the curve with equation  x equals begin inline style 1 half end style open parentheses k minus square root of k squared minus 4 y end root close parentheses.

 

10b
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5 marks

When region R is rotated 2 straight pi radians about the x-axis, the resultant solid of revolution has a volume equal to fraction numerator 1296 straight pi over denominator 5 end fraction space units cubed . 

Find the value of k.

10c
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3 marks

Use the result from part (a) to find the area of region S.

10d
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2 marks

Use your answer to part (c) to write down the area of region R.

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11
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7 marks

The diagram below shows the cross-section of a miniature goldfish bowl produced by Some Things Fishy, a specialist company supplying products for miniature goldfish enthusiasts.

q12_ib-aa-hl_adavance-integration_hard_diagram

The glass part of the bowl sits on a solid base, indicated by the shaded region on the diagram.  The cross-section of the glass part of the bowl is symmetrical about the y-axis, and may be described by the curve with equation

 x squared over 64 plus open parentheses y minus 5 close parentheses squared over 16 equals 1

The dashed horizontal line represents the diameter of the open top of the fishbowl.  All coordinates are expressed in centimetres, and for purposes of answering this question the thickness of the glass sides of the bowl may be regarded as negligible. 

Given that the diameter of the open top of the fishbowl is 15 cm, and that this is less than the diameter of the fishbowl at its widest point, find the capacity of the glass part of the fishbowl.

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1a
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3 marks

Consider the function f defined by f open parentheses x close parentheses equals open parentheses x squared minus 3 x plus 2 close parentheses open parentheses x plus 2 close parentheses comma space minus 3 less or equal than x less or equal than 5 over 4.

Find the indefinite integral

integral open parentheses x squared minus 3 x plus 2 close parentheses open parentheses x plus 2 close parentheses d x

 

1b
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4 marks

Use your answer to part (a) to calculate the area of the region enclosed by the graph of  y equals f open parentheses x close parentheses and the x-axis.

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2a
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2 marks

Find the indefinite integral for

integral sin open parentheses fraction numerator square root of 3 over denominator 2 end fraction x close parentheses d x
2b
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2 marks

Find the indefinite integral for

integral 7 over e to the power of 4 x minus 9 end exponent 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 cos open parentheses 2 open parentheses straight pi over 8 minus x close parentheses close parentheses

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3a
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2 marks

Find the indefinite integral

integral negative fraction numerator 7 over denominator 5 x end fraction d x
3b
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3 marks

Find an expression for given that

fraction numerator d y over denominator d x end fraction equals x e to the power of x squared minus 2 end exponent

and also that y equals 3 when x equals negative square root of 2

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4a
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3 marks

Find the indefinite integral for

integral open parentheses square root of 3 x end root plus fraction numerator 5 over denominator cube root of 8 x end root end fraction close parentheses d x
4b
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3 marks

Find the indefinite integral for

integral fraction numerator open parentheses 8 x close parentheses to the power of 2 over 3 end exponent minus 5 x to the power of 1 over 6 end exponent over denominator x cube root of x end fraction d x

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5a
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4 marks

Find the indefinite integral

integral fraction numerator x plus 2 x squared over denominator open parentheses 1 minus x squared close parentheses open parentheses 2 plus x squared close parentheses end fraction d x
5b
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5 marks

Let  g apostrophe open parentheses x close parentheses equals fraction numerator cos open parentheses ln space x close parentheses over denominator x end fraction for x greater than 0. 

Find  g open parentheses x close parentheses given that  g open parentheses 1 close parentheses equals straight pi.

5c
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4 marks

Show that

integral tan space x space d x equals ln open vertical bar fraction numerator 1 over denominator cos space x end fraction close vertical bar plus c

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6
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6 marks

A curve with equation y equals f open parentheses x close parentheses is such that

 fraction numerator d y over denominator d x end fraction equals fraction numerator k over denominator open parentheses 1 plus sin space pi x close parentheses open parentheses 1 minus sin space pi x close parentheses end fraction 

where k is a real constant. 

Given that the curve passes through the points open parentheses 0 comma negative 3 close parentheses and open parentheses negative 1 fourth comma negative straight pi close parentheses, find f open parentheses x close parentheses.

 

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7a
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2 marks

Explain why

fraction numerator 1 over denominator tan space theta end fraction equals fraction numerator cos space theta space space over denominator sin space theta end fraction
7b
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7 marks

Use definite integration, along with the result from part (a), to show that

integral subscript 0 superscript square root of straight pi over 6 end root end superscript fraction numerator x over denominator tan open parentheses x squared minus fraction numerator 2 straight pi over denominator 3 end fraction close parentheses end fraction d x equals negative 1 half ln open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses
7c
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1 mark

Using your knowledge of the natural logarithm function, explain (without using your GDC) why the value of the integral found in part (b) is a positive number.

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8a
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4 marks

The diagram below shows the graph of the function f which is defined by

f open parentheses x close parentheses equals x sin space 2 x comma space space space space space space space 0 less or equal than x less or equal than fraction numerator 3 straight pi over denominator 2 end fraction

mi-q8a-5-4-ib-ai-hl-further-integration-very-hard_dig

The shaded region in the diagram is the region enclosed by the x-axis and the graph of y equals f open parentheses x close parentheses.  The three sub-parts of the shaded region are denoted by R, S and T, as shown.

Find the value of

integral subscript 0 superscript fraction numerator 3 straight pi over denominator 2 end fraction end superscript x space sin space 2 x space d x

8b
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5 marks

Find the individual areas of each of the three sub-parts R, S and T of the shaded region.

8c
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4 marks

Compare the sum of the answers in part (b) to the answer in part (a) and comment on the result.

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9
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8 marks

The diagram below depicts the design for a new company logo.  The upper border of the logo is formed by a part of the curve with equation  y equals 9 minus x squared,  while the lower border of the logo is formed by a part of the curve with equation y equals 1 over 8 x squared minus 5.  As shown in the diagram, the logo is divided into a shaded part and an unshaded part by a part of the curve with equation y equals fraction numerator open parentheses 9 minus x squared close parentheses open parentheses 2 x minus 1 close parentheses over denominator 10 end fraction.

mi-q9-5-4-ib-ai-hl-further-integration-very-hard_dig

Find the percentage of the total area of the logo that is shaded.

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10a
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9 marks

The following diagram shows a part of the graph of the curve with equation y squared equals k open parentheses 1600 minus 25 open parentheses x minus 8 close parentheses squared close parentheses,  where k greater than 0 is a constant.  The point marked A is the vertex of the curve.  Region R is the region enclosed by the curve and the x-axis, for the part of the curve where y is non-negative.  Region S is the region enclosed by the curve, the positive y-axis, and the line through point A with gradient zero.

mi-q10a-5-4-ib-ai-hl-further-integration-very-hard-2_dig

When region R is rotated 2 straight pi radians about the x-axis, the resultant solid of revolution has a volume equal to fraction numerator 12800 straight pi over denominator 3 end fraction space units cubed

Find the area of region S by calculating an area between the curve and the y-axis.

10b
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4 marks

Find the area of region S by calculating an area between the curve and the x-axis.  Confirm that this matches your answer to part (a).

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11
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8 marks

The diagram below shows the cross-section of a goldfish bowl to be produced by Pieseize Manufacturing, a specialist company supplying products for goldfish enthusiasts.

 q12-5-9-ib-aa-hl-advanced-integration-very-hard-maths_diagram

The glass part of the bowl sits on a solid base, indicated by the shaded region on the diagram. The cross-section of the glass part of the bowl is symmetrical about the y-axis, and may be described by the curve with equation

x squared over 400 plus open parentheses y minus 17 close parentheses squared over 225 equals 1

The dashed horizontal line represents the diameter of the open top of the fishbowl.  The maximum depth of the fishbowl, measured along the y-axis from the diameter of the open top to where the glass part of the bowl meets the base, is indicated by d  in the diagram.  All coordinates are expressed in centimetres, and for purposes of answering this question the thickness of the glass sides of the bowl may be regarded as negligible.

The owner of the company, Skodyn Pieseize, is extremely superstitious and is obsessed with the number 23. Therefore he insists that the capacity of the glass part of this new fishbowl must be exactly 23 litres.  Find the value of d that satisfies this requirement.

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