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

Topic Questions

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Medium

4 marks

$f (x) = \frac{x + 2}{x^{2} + 9}$

(a)

Show that

$\int f (x) d x = A \ln (x^{2} + 9) + B \arctan (\frac{x}{3}) + c$

where

c

is an arbitrary constant and

A

and

B

are constants to be found.

How did you do?

3 marks

(b)

Hence show that the mean value of

f (x)

over the interval [0, 3] is

$\frac{1}{6} \ln 2 + \frac{1}{18} π$

How did you do?

2 marks

(c)

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

$f (x) + \ln k$

where

k

is a positive constant, giving your answer in the form

p + \frac{1}{6} \ln q,

where

p

and

q

are constants and

q

is in terms of

k

How did you do?

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