Methods in Calculus (Edexcel A Level Further Maths: Core Pure)

Exam Questions

39 mins5 questions
1a
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4 marks

space f left parenthesis x right parenthesis equals fraction numerator x plus 2 over denominator x squared plus 9 end fraction

(a)
Show that

integral f left parenthesis x right parenthesis straight d x italic space italic equals A ln left parenthesis x squared space plus space 9 right parenthesis italic space italic plus italic space B arctan open parentheses x over 3 close parentheses plus straight c

where c is an arbitrary constant and A and B are constants to be found.
1b
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3 marks
(b)
Hence show that the mean value of space f left parenthesis x right parenthesis over the interval [0, 3] is

1 over 6 ln 2 plus 1 over 18 straight pi

1c
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2 marks
(c)
Use the answer to part (b) to find the mean value, over the interval [0, 3], of

space f left parenthesis x right parenthesis space plus space ln k

where k is a positive constant, giving your answer in the form p space plus space 1 over 6 ln space q comma wherespace p and q are constants and q is in terms of k.

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2a
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1 mark
a)
Explain whyintegral subscript 1 superscript infinity fraction numerator 1 over denominator x left parenthesis 2 x plus 5 right parenthesis end fraction straight d x is an improper integral.
2b
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6 marks
(b)
Prove that

integral subscript 1 superscript infinity fraction numerator 1 over denominator x left parenthesis 2 x plus 5 right parenthesis end fraction straight d x equals a ln b

where a and b are rational numbers to be determined.

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3a
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3 marks
(a)

y space equals space tan to the power of negative 1 end exponent space x

Assuming the derivative of tan space x comma prove that

fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator 1 plus x squared end fraction

3b
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5 marks

space f left parenthesis space x space right parenthesis space equals space x space tan to the power of negative 1 end exponent space 4 x

(b)
Show that

integral f left parenthesis x right parenthesis straight d x equals A x squared space tan to the power of negative 1 end exponent space 4 x space plus B x space plus space C space tan to the power of negative 1 end exponent space 4 x plus space k

where k is an arbitrary constant and A comma space B and C are constants to be determined.
3c
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2 marks
(c)
Hence find, in exact form, the mean value of space f left parenthesis x right parenthesis over the interval open square brackets 0 comma space fraction numerator square root of 3 over denominator 4 end fraction close square brackets

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4
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7 marks

Show that

integral subscript 0 superscript infinity fraction numerator 8 x minus 12 over denominator left parenthesis 2 x squared plus 3 right parenthesis left parenthesis x plus 1 right parenthesis end fraction straight d x equals ln k

where k is a rational number to be found

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5a
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4 marks

space f left parenthesis x right parenthesis equals fraction numerator 1 over denominator square root of 4 x squared plus 9 end root end fraction

(a)
Using a substitution, that should be stated clearly, show that

integral f space left parenthesis x right parenthesis straight d x space equals space A space sinh to the power of negative 1 end exponent space left parenthesis B x right parenthesis space plus space c

where c is an arbitrary constant and A and B are constants to be found.
5b
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2 marks
(b)
Hence find, in exact form in terms of natural logarithms, the mean value ofspace f left parenthesis x right parenthesis over the interval [0, 3].

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