Further Integration (Edexcel International A Level Maths: Pure 3)

Exam Questions

4 hours39 questions
1
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6 marks
(i)
Find the integral
        integral space 1 over x straight d x
(ii)

Use calculus to evaluate

      integral subscript 0 superscript 1 space e to the power of x space straight d x.

(iii)
Find an expression for y given that
      y space equals integral 3 space cos space theta space space straight d theta.

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2
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8 marks
(i)

Integrate

  integral space 8 left parenthesis 2 x minus 1 right parenthesis cubed space straight d x.

(ii)

Use calculus to find the exact value of

   integral subscript 0 superscript straight pi over 4 end superscript space sin space 2 x space straight d x.

(iii)

Find an expression for y given that

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

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3a
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2 marks

Given the identity cos space 2 A identical to 1 minus 2 sin squared space A, show that

         sin squared space A space equals 1 half open parentheses 1 minus cos space 2 A close parentheses.

3b
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3 marks

Hence find the exact value of

            integral subscript straight pi over 2 end subscript superscript straight pi space sin squared x space straight d x.

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4
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4 marks
(i)
Given that  straight f left parenthesis x right parenthesis equals 2 x squared plus 5, find straight f to the power of apostrophe left parenthesis x right parenthesis
(ii)
Hence, or otherwise, find
      integral fraction numerator 4 x over denominator 2 x squared plus 5 end fraction straight d x.

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5
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5 marks

Find the exact value of

         integral subscript 1 superscript 2 e to the power of 3 x plus 2 end exponent space straight d x.

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6
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8 marks
(i)

Integrate

   integral sin space left parenthesis 2 x plus 1 right parenthesis space space space d x.

(ii)

Use calculus to find the exact value of

   integral subscript straight pi over 6 end subscript superscript straight pi over 4 end superscript space sec squared open parentheses 2 theta minus straight pi over 6 close parentheses space straight d theta.

(iii)

Find an expression for y given that

   fraction numerator straight d y over denominator straight d x end fraction equals 4 space cos open parentheses 5 x plus 2 close parentheses.

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7
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4 marks
(i)

Find an expression for y given that

   begin mathsize 20px style fraction numerator straight d y over denominator straight d x end fraction equals 7 e to the power of 3 x minus 4 end exponent end style.

(ii)
Integrate
   integral space fraction numerator 1 over denominator 4 x plus 9 end fraction space straight d x.

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8
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5 marks
(i)
Integrate
   integral space fraction numerator 1 over denominator 2 minus 3 x end fraction space straight d x.
(ii)

Find an expression for y given that

   begin mathsize 20px style y equals integral space open parentheses fraction numerator 2 over denominator 2 x plus 1 end fraction minus fraction numerator 1 over denominator 3 minus x end fraction close parentheses space straight d x end style

giving your answer as a single logarithm.

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This topic pack contains questions on the following:

  • Integrating Other Functions (Trig, ln & e etc)

  • Reverse Chain Rule

  • f'(x)/f(x)

  • Integrating with Trigonometric Identities

1a
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1 mark

Find the integral

         integral space sin space x space straight d x

1b
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3 marks

Use calculus to evaluate

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

1c
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2 marks

Find an expression for y given that

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

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2a
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2 marks

Integrate

      integral space cos space 2 x space straight d x

2b
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4 marks

Use calculus to find the value of

            integral subscript 0 superscript 2 space 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

               begin mathsize 20px style fraction numerator straight d y over denominator straight d x end fraction equals e to the power of 5 x end exponent end style

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3
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6 marks

Given that

        cos space 2 theta identical to 2 cos squared space theta 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 space space cos squared space theta space straight d theta

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4
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3 marks

Find

      integral space square root of 1 plus space cot squared x end root space straight d x

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5
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4 marks
(i)

Given that begin mathsize 20px style straight f left parenthesis x right parenthesis equals 2 x cubed plus 4 x end style,  find straight f to the power of apostrophe left parenthesis x right parenthesis .

(ii)
Hence, or otherwise, find
            begin mathsize 20px style integral fraction numerator 3 x squared plus 2 over denominator 2 x cubed plus 4 x end fraction space straight d x end style

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6a
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2 marks

Integrate

      integral fraction numerator 1 over denominator 2 x plus 3 end fraction space straight d x

6b
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2 marks

Find an expression for given that

               fraction numerator straight d y over denominator straight d x end fraction equals 3 e to the power of 3 x plus 1 end exponent

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7a
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2 marks

Find an expression for straight f open parentheses x close parentheses given that

               straight f open parentheses x close parentheses equals integral space sin open parentheses 2 x plus 1 close parentheses space straight d x

7b
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2 marks

Find an expression for given that

            fraction numerator straight d y over denominator straight d x end fraction equals space 3 space cos open parentheses 3 x plus 5 close parentheses

7c
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3 marks

Use calculus to find the exact value of

            integral subscript fraction numerator 7 straight pi over denominator 6 end fraction end subscript superscript fraction numerator 4 straight pi over denominator 3 end fraction end superscript space sec squared open parentheses x minus straight pi close parentheses space straight d x

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8a
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2 marks

Use the identity

         sin space 2 A identical to 2 space sin A space space cos A

to show that

         4 space sin theta over 2 cos theta over 2 identical to 2 space sin space theta

8b
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3 marks

Hence, or otherwise, find the integral

               integral space 4 space sin theta over 2 cos theta over 2 space straight d theta

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9
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3 marks

Show that

      integral subscript 2 superscript 5 space 4 e to the power of 2 x minus 4 end exponent space straight d x space equals space 2 open parentheses e to the power of 6 space minus 1 close parentheses

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10a
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4 marks

Given that the graph of  begin mathsize 20px style y equals straight f left parenthesis x right parenthesis end style passes through the point left parenthesis 0 comma 4 right parenthesis and that

               straight f apostrophe open parentheses x close parentheses equals fraction numerator 12 x squared plus 7 over denominator 2 open parentheses 4 x cubed plus 7 x plus 4 close parentheses end fraction

Find straight f left parenthesis x right parenthesis

10b
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1 mark

Explain why x equals negative 1 half must be excluded from the domain of straight f open parentheses x close parentheses.

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11
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4 marks

Show that

      integral tan space x space space straight d x equals ln vertical line s e c space x vertical line plus c space

where c is a constant.

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1a
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2 marks

Find the integral

         integral space 5 open parentheses e to the power of 5 x end exponent minus e to the power of negative 5 x end exponent close parentheses space straight d x

1b
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2 marks

Find an expression for y given that

            y equals integral left parenthesis sin space x plus cos space x right parenthesis space space straight d x space

1c
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3 marks

Use calculus to evaluate

         integral subscript negative 8 end subscript superscript negative 2 end superscript space 1 over x straight d x

giving your answer as a single logarithm.

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2a
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2 marks

Integrate

        integral 2 space sin space x space space cos space x space space straight d x

2b
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4 marks

Use calculus to find the value of

            integral subscript 1 superscript 3 left parenthesis 4 x plus 1 right parenthesis to the power of 5 space straight d x

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3a
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3 marks

Show that, for theta not equal to k pi(where k is an integer),

            fraction numerator 2 minus 2 space cos squared theta over denominator sin space 2 theta end fraction equals tan space theta

3b
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4 marks

Use calculus and your result from part [a] to show that

          begin mathsize 20px style integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript space fraction numerator 2 minus 2 space cos to the power of 2 space end exponent theta over denominator sin space 2 theta end fraction space straight d theta equals 1 half In space 3 end style

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4
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6 marks

Use calculus to find the exact value of

            integral subscript straight pi over 2 end subscript superscript fraction numerator 5 straight pi over denominator 6 end fraction end superscript space space fraction numerator 2 space cos space x over denominator 1 minus cos space 2 x end fraction space straight d x.

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5
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5 marks

Use calculus to show that

            integral subscript 0 superscript 1 space fraction numerator 3 x e to the power of negative 3 x squared end exponent over denominator 3 minus 2 e to the power of negative 3 x squared end exponent end fraction space straight d x space equals 1 fourth In open parentheses 3 minus 2 e to the power of negative 3 end exponent close parentheses.

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6a
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2 marks

Find an expression for straight f open parentheses x close parentheses given that

            straight f open parentheses x close parentheses equals integral fraction numerator 5 over denominator 2 x plus 3 end fraction space straight d x

6b
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3 marks

Find an expression for y given that

               fraction numerator straight d y over denominator straight d x end fraction equals 2 e to the power of 3 x minus 5 end exponent minus 5 e to the power of 5 x end exponent

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7a
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2 marks

Show that

      sin squared space open parentheses 3 x plus 2 close parentheses identical to 1 half open parentheses 1 minus cos open parentheses 6 x plus 4 close parentheses close parentheses

7b
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3 marks

Hence, or otherwise, find an expression for

               integral sin squared left parenthesis 3 x plus 2 right parenthesis space straight d x

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8a
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2 marks

Write begin mathsize 20px style e to the power of 2 x end exponent left parenthesis 1 plus e to the power of 2 x end exponent plus e squared right parenthesis end style in the form   begin mathsize 20px style e to the power of straight f left parenthesis x right parenthesis plus e to the power of g left parenthesis x right parenthesis plus e to the power of h left parenthesis x right parenthesis end style,  where straight f left parenthesis x right parenthesis comma g left parenthesis x right parenthesis space space and space h left parenthesis x right parenthesis are all linear functions of x.

8b
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4 marks

Hence show that

         integral subscript 0 superscript 1 space e to the power of 2 x end exponent open parentheses 1 plus e to the power of 2 x end exponent plus e squared close parentheses space straight d x space equals 3 over 4 open parentheses e to the power of 4 minus 1 close parentheses

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9
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4 marks

Given that straight f to the power of apostrophe left parenthesis x right parenthesis equals 2 tan space 3 x spaceand that straight f open parentheses 0 close parentheses equals 2 over 3 show that

         straight f open parentheses x close parentheses equals 2 over 3 open parentheses 1 plus ln vertical line sec space 3 x vertical line close parentheses

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10
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5 marks

Given

      straight f open parentheses x close parentheses equals integral fraction numerator 3 open parentheses 8 x cubed plus 3 x close parentheses over denominator 2 open parentheses 4 x to the power of 4 plus 3 x squared plus 5 close parentheses end fraction space straight d x space space space space space space space space space space x element of straight real numbers

and that the graph of  y equals straight f left parenthesis x right parenthesis passes through the point left parenthesis 1 comma   ln space 144 right parenthesis find straight f open parentheses x close parentheses.

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1a
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3 marks

Find an expression for y given that

               fraction numerator straight d y over denominator straight d x end fraction equals space 5 space cos squared space 4 x space sin space 4 x

1b
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3 marks

Integrate

      integral 3 x left parenthesis 5 x squared plus 4 right parenthesis to the power of 4 space straight d x

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2
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5 marks

Find

   integral left parenthesis 2 space tan space x plus 3 space sec space x right parenthesis to the power of 2 space end exponent straight d x

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3a
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3 marks

Show that

      integral space tan space k x space straight d x equals fraction numerator 1 over denominator k space end fraction space ln vertical line sec space k x vertical line plus c

where k is a constant, and c is the constant of integration.

3b
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5 marks

Use calculus to find the exact value of

         integral subscript straight pi over 18 end subscript superscript straight pi over 9 end superscript space fraction numerator cosec to the power of 2 space end exponent 3 theta over denominator 3 space cot space 3 theta end fraction space straight d theta

writing your answer in the form   a space ln space b, where a and b are rational numbers to be found.

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4
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5 marks

Use calculus to find the value of

         integral subscript 2 superscript 6 space open parentheses 1 over x plus fraction numerator 4 over denominator 2 x end fraction plus fraction numerator 3 over denominator 3 x minus 2 end fraction close parentheses space straight d x

giving your answer in the form   p space ln space p plus q space ln space q space,  where p and q are prime numbers to be found.

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5a
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3 marks

Show that

      open parentheses cos space open parentheses theta plus straight pi over 8 close parentheses plus sin space open parentheses theta plus straight pi over 8 close parentheses close parentheses open parentheses sin space open parentheses theta plus straight pi over 8 close parentheses minus cos space open parentheses theta plus straight pi over 8 close parentheses close parentheses identical to negative cos space open parentheses 2 theta plus straight pi over 4 close parentheses

5b
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3 marks

Hence, or otherwise, find the exact value of

         integral subscript 0 superscript straight pi over 8 end superscript open parentheses sin squared space open parentheses theta plus straight pi over 8 close parentheses minus cos to the power of 2 space end exponent open parentheses theta plus straight pi over 8 close parentheses close parentheses space straight d theta

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6a
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3 marks

Show that

      integral subscript 0 superscript 1 space e to the power of a x plus b end exponent space straight d x space equals space e to the power of b open parentheses fraction numerator e to the power of a minus 1 over denominator a end fraction close parentheses

where a and b are constants, and a not equal to 0.

6b
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2 marks

Using your working from part [a], or otherwise, evaluate 

            integral subscript 0 superscript c space e to the power of a x plus b end exponent space straight d x

giving your answer in terms of  a comma b space and space c, where a comma b space and space straight c are constants, and a not equal to 0.

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7a
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3 marks

Show that

      open parentheses 1 plus space cot space open parentheses 2 theta plus straight pi over 4 close parentheses close parentheses open parentheses 1 minus cot space open parentheses 2 theta plus straight pi over 4 close parentheses close parentheses identical to 2 minus space cosec squared space open parentheses 2 theta space plus straight pi over 4 close parentheses

7b
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3 marks

Hence, or otherwise, find an expression for straight f open parentheses theta close parentheses given that

            straight f apostrophe open parentheses theta close parentheses space equals space open parentheses 2 plus space 2 space cot space open parentheses 2 straight theta plus straight pi over 4 close parentheses close parentheses open parentheses 2 minus 2 space cot space open parentheses 2 straight theta space plus straight pi over 4 close parentheses close parentheses

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8
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7 marks

Show that there are no positive values of a and b that satisfy the equation

               integral subscript 0 superscript 1 space fraction numerator 2 a over denominator a x plus b end fraction space straight d x space equals integral subscript b superscript 2 a end superscript space 1 over x space straight d x

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9
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6 marks

Given that

      straight f apostrophe open parentheses straight theta close parentheses equals fraction numerator 9 over denominator 3 minus 3 space sin squared space open parentheses 3 theta minus pi over 6 close parentheses end fraction

and also that

         open square brackets straight f open parentheses straight pi over 6 close parentheses close square brackets squared minus open square brackets straight f open parentheses straight pi over 9 close parentheses close square brackets squared space equals 8 over 3 open parentheses 1 minus square root of 3 close parentheses

find straight f open parentheses theta close parentheses.

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10
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6 marks

Given that  straight f apostrophe open parentheses x close parentheses equals 1 fourth space tan space open parentheses x over 2 close parentheses space and space straight f open parentheses fraction numerator 2 straight pi over denominator 3 end fraction close parentheses equals 1 half In space 6, show that 

               straight f open parentheses x close parentheses equals In space square root of open vertical bar 3 space sec open parentheses x over 2 close parentheses close vertical bar end root           

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