### Question

Let f(x) $$\frac{6x^{2}+8x+9}{(2-x)(3+2x)^{2}}$$

(i) Express f(x) in partial fractions.

(ii) Hence, showing all necessary working, show that $$\int_{-1}^{0}f(x)dx=1+\frac{1}{2}In (\frac{3}{4}).$$

(i) State or imply the form

Use a correct method to find a constant

Obtain one of A = 1, B = – 1, C = 3

Obtain a second value

Obtain the third value

[Mark the form $$\frac{A}{2-x}+\frac{Dx+E}{(3+2x)^{2}}$$, where A = 1, D = – 2 and E = 0, B1M1A1A1A1 as above.]

(ii) Integrate and obtain terms$$-In(2-x)-\frac{1}{2} In (3+2x)-\frac{3}{2(3+2x)^{2}}$$

Substitute correctly in an integral with terms a ln (2 – x),

b ln (3 + 2x) and c / (3 + 2x) where abc ≠ 0

Obtain the given answer after full and correct working
[Correct integration of the

A, D, E form gives an extra constant term if integration by

parts is used for the second partial fraction.]

### Question

Let $$fx =\frac{x^{2}+x+6}{x^{2}(x+2)}$$

(i) Express fx in partial fractions.

(ii) Hence, showing full working, show that the exact value of $$\int _{1}^{4f(x)dx is \frac{9}{4}}$$<

(i) State or imply the form $$\frac{A}{x}+\frac{B}){x^{2}}+\frac{c}{x+2}\ Use a correct method for finding a constant Obtain one of A = – 1, B = 3, C = 2 Obtain a second value Obtain the third value (ii) Integrate and obtain terms In \( x-\frac{3}{x}+2 In(x+2)$$ Substitute limits correctly in an integral with terms a In x$$\frac{b}{x}$$ and c x ln 2 ( + ) , where abc
≠ 0

Obtain$$\frac{9}{4}$$ following full and exact working

### Question

Let f(x )=$$\frac{16-17x}{(2+x)(3-x)^{2}}$$

(i) Express f(x )in partial fractions.

(ii) Hence obtain the expansion of f(x )in ascending powers of x, up to and including the term in $$x^{2}$$

(i)State or imply the form $$\frac{A}{2+x}+\frac{B}{3-x}+\frac{C}{\left ( 3-x \right )^{2}}$$

Use a correct method to obtain a constant

Obtain one of A=2,B=2,C=-7

Obtain a second value

Obtain the third value

(ii) Use a correct method to find the first two terms of the expansion of $$\left ( 2+x \right )^{-1},\left ( 3-x \right )^{-1}$$ or $$\left ( 3-x \right )^{-2}$$, or equibvalent ,e.g.$$\left ( 1+\frac{1}{2}x \right )^{-1}$$

Obtain correct Unsimplified expansions up to the term in$$x^{2}$$ of each partial fraction.

Obtain final answer $$\frac{8}{9}-\frac{43}{54}x+\frac{7}{108}x^{2}$$

### Question

Let f(x) $$\frac{12+12x-4x^{2}}{(2+x)(3-2x)}$$.

(i) Express f(x )in partial fractions.

(ii) Hence obtain the expansion of f(x )in ascending powers of x, up to and including the term in $$x^{2}$$

(i) State or imply the form$$\frac{5}{2}+\frac{1}{2}i$$

Use a correct method for finding a constant

Obtain one of A = 2, B = – 4 and C = 6

Obtain a second value

Obtain the third value

(ii) Use correct method to find the first two terms of the expansion of$$(2+x)^{-1} or (3-2x)^{-1}$$, or equivalent

Obtain correct unsimplified expansions up to the term in $$x^{2}$$of each partial fraction

Add the value of A to the sum of the expansions

Obtain final answer $$2+\frac{7}{3}x+\frac{7}{18}x^{2}$$

### Question

(i) Express $$\frac{4+12x+x^{2}}{(3-x)(1+2x)^{2}}$$in partial fractions.

(ii) Hence obtain the expansion of $$\frac{4+12x+x^{2}}{(3-x)(1+2x)^{2}}$$ in ascending powers of x, up to and including the term in$$x^{2}$$

(i) Either State or imply partial fractions are of form $$\frac{A}{3-x}+\frac{B}{1+2x}+\frac{C}{(1+2x)^{2}}$$

Use any relevant method to obtain a constant

Obtain A = 1

Obtain B=$$\frac{3}{2}$$

Obtain C=$$-\frac{1}{2}$$

Or State or imply partial fractions are of form $$\frac{A}{3-x}+\frac{Dx+E}{(1+2x)^{2}}$$

Use any relevant method to obtain a constant

Obtain A = 1

Obtain D = 3

Obtain E = 1

(ii) Obtain the first two terms of one of the expansion$$(3-x)^{-1},\left ( 1-\frac{1}{3}x \right )^{-1} (1+2x)^{-1} and (1+2x)^{-2}$$

Obtain correct unsimplified expansion up to the term in $$x^{2}$$ of each partial fraction, following in each case the value of A, B, C

Obtain answer $$\frac{4}{3}-\frac{8}{9}x+\frac{1}{27}x^{2}$$

[If A, D, E approach used in part (i), give M1 A1  A1 for the expansions,  for
multiplying out fully and  for final answer]

### Question

Let $$f\left ( x \right ) = \frac{2x^{2}-7x-1}{\left ( x-2 \right )\left ( x^{2}+3 \right )}$$.

(i) Express fx in partial fractions. [5]

(ii) Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x2.[5]

Ans:

(i) State or imply partial fractions are of the form $$\frac{A}{x-2}+\frac{Bx+C}{x^{2}+3}$$
Use a relevant method to determine a constant
Obtain one of the values A = –1, B = 3, C = –1
Obtain a second value
Obtain the third value

(ii) Use correct method to obtain the first two terms of the expansions of $$\left ( x-2 \right )^{-1},\left ( 1-\frac{1}{2}x \right )^{-1},\left ( x^{2}+3\right )^{-1} or \left ( 1+\frac{1}{3}x^{2} \right )^{-1}$$
Substitute correct unsimplified expansions up to the term in x2 into each partial fraction
Multiply out fully by Bx + C, where BC ≠ 0
Obtain final answer $$\frac{1}{6}+\frac{5}{4}x+\frac{17}{72}x^{2}$$ , or equivalent
[Symbolic binomial coefficients, e.g. $$\begin{pmatrix}-1\\1\\\end{pmatrix}$$ are not sufficient for the M1. The f.t. is on A, B, C.]
[In the case of an attempt to expand $$\left ( 2x^{2}-7x-1 \right )\left ( x-2 \right )^{-1}\left ( x^{2}+3 \right )^{-1}$$ , give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.]
[If B or C omitted from the form of partial fractions, give B0M1A0A0A0 in (i); M1A1 A1 in (ii)]

### Question

Let $$f(x)=\frac{4x^{2}+9x-8}{(x+2)(2x-1)}$$

(i) Express f(x) in the form $$A+\frac{B}{x+2}+\frac{C}{2x-1}$$

(ii) Hence show that$$\int_{1}^{4}f(x)dx=6+\frac{1}{2}ln\left ( \frac{16}{7} \right )$$

8(i) Use a relevant method to determine a constant
Obtain one of the values A = 2, B = 2, C = –1
Obtain a second value
Obtain the third value
8(ii) Integrate and obtain terms $$2x+2ln(x+2)-\frac{1}{2}ln(2x-1)$$ (deduct B1 for each error or omission) [The FT is on A, B and C]

Substitute limits correctly in an integral containing terms a x ln( 2) + and ln(2 1) b x − ,
where ab ≠ 0 Use at least one law of logarithms correctly D Obtain the given answer after full and correct working

### Question

(a) Show that $$\int_{2}^{4}4xlnxdx=56ln2-12$$. [5]

(b) Use the substitution u = sin 4x to find the exact value of $$\int_{0}^{\frac{1}{24}\pi }cos^{3}4xdx.$$ [5]

Ans:

(a) Carry out integration by parts and reach $$ax^{2}lnx+b\int \frac{1}{2}x^{2}dx$$
Obtain $$2x^{2}lnx-\int \frac{1}{x}.2x^{2}dx$$
Obtain $$2x^{2}lnx-x^{2}$$
Use limits, having integrated twice
Confirm given result 56 ln 2 – 12

(b) State or imply $$\frac{du}{dx}=4cos4x$$
Carry out complete substitution except limits
Obtain $$\int \left ( \frac{1}{4}- \frac{1}{4}u^{2}\right )du$$ or equivalent
Integrate to obtain form k1u + k2u3 with non-zero constants k1, k2
Use appropriate limits to obtain $$\frac{11}{96}$$

### Question

For each of the following curves, find the gradient at the point where the curve crosses the y-axis:

(i) $$y=\frac{1+x^{2}}{x+e^{2x}};$$ [3]

(ii) $$2x^{3}+5xy+y^{3}=8$$ [4]

Ans:

(i) Use correct quotient rule or equivalent
Obtain $$\frac{\left ( 1+e^{2x} \right )2x-\left ( 1+x^{2} \right )2e^{2x}}{\left ( 1+e ^{2x}\right )^{2}}$$ or equivalent
Substitute x = 0 and obtain $$-\frac{1}{2}$$ or equivalent
(ii) Differentiate y3 and obtain $$3y^{2}\frac{dy}{dx}$$
Differentiate 5xy and obtain $$5y+5x\frac{dy}{dx}$$
Obtain $$6x^{2}+5y+5x\frac{dy}{dx}+3y^{2}\frac{dy}{dx}=0$$
Substitute x = 0, y = 2 to obtain $$-\frac{5}{6}$$ or equivalent following correct work

### Question

(i) Express $$\frac{1}{x}(2x + 3)$$ in partial fractions.

(ii) The variables x and y satisfy the differential equation

$$x(2x+3)\frac{dy}{dx}=y$$.

and it is given that y = 1 when x = 1. Solve the differential equation and calculate the value of y when x = 9, giving your answer correct to 3 significant figures.

9(i) Carry out a relevant method to obtain A and B such that $$\frac{1}{x(2x+3)}\equiv \frac{A}{x}+\frac{B}{2x+3},$$or

equivalent

Obtain   A =$$\frac{1}{3}$$ and   B = $$−\frac{2}{3}$$ , or equivalent

9(ii) Separate variables and integrate one side
Obtain term ln  y
Integrate and obtain terms $$\frac{1}{3}lnx -\frac{1}{3}ln(2x+3,)$$

Use x = 1 and y = 1 to evaluate a constant, or as limits, in a solution containing

Obtain correct solution in any form, e.g $$\frac{1}{3}lnx -\frac{1}{3}ln(2x+3,)$$
Obtain answer y = 1.29 (3s.f. only)

### Question

Let f(x)=$$\frac{x(6-x)}{(2+x)(4+x^{2})}$$

(i) Express f(x )in partial fractions.

(ii) Hence obtain the expansion of f(x )in ascending powers of x, up to and including the term in $$x^{2}$$

(i) State or imply the form  $$\frac{A}{2+x}+\frac{Bx+C}{4+x^{2}}$$

Use a relevant method to determine a constant

Obtain one of the values A = – 2, B = 1, C = 4

Obtain a second value

Obtain the third value

### Question

Let $$f(x)=\frac{8+5x+12x^{2}}{(1-x)(2+3x)^{2}}$$   .
(a) Express (fx) in partial fractions.

(b) Hence obtain the expansion of f (x) in ascending powers of x, up to and including the term in x2.         [5]

(a) State or imply the form $$\frac{A}{1-x}+\frac{B}{2+3x}+\frac{C}{(2+3x)^{2}}$$

Use a correct method for finding a coefficient
Obtain one of A = 1, B = –1 , C = 6
Obtain a second value
Obtain the third value

(b) Use a correct method to find the first two terms of the expansion

of  $$(1-x)^{1}, (2+3x)^{-1}, \left ( 1+\frac{3}{2}x \right )^{-1}, (2+3x)^{-2}\ or \ \left ( 1+\frac{3}{2}x \right )^{-2}$$

Obtain correct un-simplified expansions up to the term in
of each partial fraction

Obtain final answer $$2-\frac{11}{4}x+10x^{2}, or\ equivalent$$

### Question

Let $$f(x)=\frac{\cos x}{1+\sin x}$$

(a) Show that f′x < 0 for all x in the interval $$-\frac{1}{2}\pi < x< \frac{3}{2}\pi$$

(b) Find $$\int_{\frac{1}{6}\pi }^{\frac{1}{2}\pi }f(x)dx$$. Give your answer in a simplified exact form.

Ans

(a) Use quotient or product rule
Obtain derivative in any correct form e.g $$\frac{-\sin x(1+\sin x)-\cos x(\cos x)}{(1+\sin x)^{2}}$$

Use Pythagoras to simplify the derivative
Justify the given statement

(b) State integral of the form a ln (1 + sin x)
State correct integral ln (1 + sin x) A1
Use limits correctly

Obtain answer $$ln\frac{4}{3}$$

### Question

Let $$f(x)=\frac{2+11x-10x^2}{(1+2x)(1-2x)(2+x)$$
(a) Express f(x) in partial fractions.
(b) Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in $$x^2$$.

Ans:

(a) State or imply the form $$\frac{A}{1+2x}+\frac{B}{1-2x}+\frac{C}{2+x}$$
Use a correct method for finding a constant
Obtain one of A = -2, B = 1 and C = 4
Obtain a second value
Obtain the third value
(b) Use correct method to find the first two terms of the expansion of $$(1+2x)^{-1}$$.
$$(1-2x)^{-1}.(2+x)^{-1}or (1+\frac{1}{2}x)^{-1}$$
Obtain correct unsimplified expansions up to the terms in $$x^2$$ of each partial fraction
Obtain final answer $$1+5x-\frac{7}{2}x^2$$

### Question

Find $$\int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi}x sec^2xdx$$. Give your answer in a simplified exact form.

Ans:

Integrate by parts and reach axtanx + b$$\int$$tan x dx
Obtain x tan x – $$\int$$ tan x dx
Complete the integration, obtaining a term $$\pm In cosx$$, or equivalent
Obtain integral xtan x + In cos x, or equivalent
Substitute limits correctly, having integrated twice
Use a law of logarithms
Obtain answer $$\frac{5}{18} \sqrt{3 \pi} – \frac{1}{2} In 3$$, or exact simplified equivalent

### Question

The variables x and t satisfy the differential equation $$\frac{dx}{dt}=x^{2}(1+2x),\ and\ x=1 \ when\ t=0$$

Using partial fractions, solve the differential equation, obtaining an expression for t in terms of x.                        [11]

Ans

State a suitable form of partial fractions for  $$\frac{1}{x^{2}(1+2x)}$$

Use a relevant method to determine a constant

Obtain one of A = – 2, B = 1 and C = 4

Obtain a second value
Obtain the third value
Separate variables correctly and integrate at least one term

Obtain terms $$-2\ln\ x-\frac{1}{x}+ 2ln(l+2x) \ and \ t$$

Evaluate a constant, or use limits x = 1, t = 0 in a solution containing terms t,
a ln x and b  ln  (1 + 2x ), where ab≠0

Obtain a correct expression for t in any form, e.g  $$t=-\frac{1}{x}+2 ln\left ( \frac{1+2x}{3x} \right )+1$$

### Question

(a) Prove that  $$\frac{1-\cos 2\theta }{1+\cos 2\theta }=\tan ^{2}\theta$$                                                                               [2]

(b) Hence find the exact value of $$\int_{\frac{1}{6}\pi }^{\frac{1}{3}\pi } \frac{1-\cos 2\theta }{1+\cos 2\theta }$$    [4]

Ans

4 (a) Use correct double angle formula or t-substitution twice

Obtain $$\frac{1-\cos 2\theta }{1+\cos 2\theta }=tan^{2}\theta$$  from correct working

4 (b) Express 2 tanθ in terms of 2 secθ

Integrate and obtain terms tanθ – θ

Substitute limits correctly in an integral of the form a  tanθ + bθ  , where ab≠0

Obtain answer $$\frac{2}{3}\sqrt{3}-\frac{1}{6}\pi$$

### Question

Let $$f(x) = \frac{5a}{(2x-a)(3a-x)}$$, where a is a positive constant.
(a) Express f(x) in partial fractions.
(b) Hence show that $$\int_{a}^{2a}f(x)dx=In6$$.

Ans:

1. Carry out a relevant method to determine constants A and B such that
$$\frac{5a}{(2x-a)(3a-x)}=\frac{A}{2x-a}+\frac{B}{3a-x}$$
Obtain A=2
Obtain B=1
2. Integrate and obtain terms In(2x-a)-In(3a-x)
Substitute limits correctly in a solution containing terms of the form bIn(2x-a) and cIn(3a-x), where bc≠0
Obtain the given answer showing full and correct working

### Question

(i) Prove the identity tan $$tan2\Theta -tan\Theta sec2\Theta$$

(ii) Hence show that $$\int_{0}^{\frac{1}{6}\pi }tan\Theta sec2\Theta d\Theta =\frac{1}{2}In\frac{3}{2}$$

(i) EITHER: Use tan 2A formula to express LHS in terms of tanθ Express as a single fraction in any correct form Use Pythagoras or cos 2A formula Obtain the given result correctly

OR: Express LHS in terms of sin 2θ, cos 2θ, sin θ and cosθ
Express as a single fraction in any correct form
Use Pythagoras or cos 2A formula or sin(A – B) formula Obtain the given result correctly

(ii) Integrate and obtain a term of the form aln(cos2 ) θ or bln(cos ) θ (or secant equivalents)

Obtain integral $$-\frac{1}{2}$$ ln(cos 2 θ ) ln(cos θ )   , or equivalent

Substitute limits correctly (expect to see use of both limits)
Obtain the given answer following full and correct working

### Question

Let f(x )= $$\frac{4x^{2}+12}{(x+1)(x-3)^{2}}$$

(i) Express f(x )in partial fractions.

(ii) Hence obtain the expansion of f(x )in ascending powers of x, up to and including the term in

(i) State or imply the form

Use a correct method to determine a constant M1
Obtain one of the values A = 1, B = 3, C = 12  Obtain a second value
Obtain a third value

[Mark the form where A =1, D = 3, E = 3, B1 M1 A1 A1 A1 as above.]

(ii) Use correct method to find the first two terms of the expansion of $$(x+1)^{-1},(x-3)^{-1},(1-\frac{1}{3}x)^{-1}$$ \

$$(x-3)^{-2},(1-\frac{1}{3}x)^{-2}$$

Obtain correct unsimplified expansions up to the term in$$x^{2}$$

of each partial fraction

Obtain final answer$$\frac{4}{3}-\frac{4}{9}x+\frac{4}{3}x^{2}$$ , or equivalent

### Question

(a)Find the quotient and remainder when 8x3 + 4x2 + 2x + 7 is divided by 4x2 + 1.

(b) Hence find the exact value of $$\int_{0}^{\frac{1}{2}} \frac{8x^{3}+4x^{2}+2x+7}{4x^{2}+1}dx.$$

Ans:

(a)Commence division and reach quotient of the form 2x ± 1

Obtain (quotient) 2x + 1

Obtain (remainder) 6

(b)

Obtain terms x2+x

Obtain term of the form a tan-1 2x

Obtain term 3tan-1 2x

Use x = 0 and x = $$\frac{1}{2}$$ as limits in a solution containing a term of the form a tan-1 2x

Obtain final answer $$\frac{3}{4}\left ( 1+\pi \right )$$ , or exact equivalent

### Question

Let f(x) = $$\frac{3x^{3}+6x-8}{x\left ( x^{^{2}}+2 \right )}$$ .

(i) Express f(x) in the form $$A+\frac{B}{x}+\frac{Cx+D}{x^{2}+2}$$ .[5]

(ii) Show that $$\int_{1}^{2}f\left ( x \right )dx = 3 – ln 4$$ .[5]

Ans:

(i) State or obtain A = 3
Use a relevant method to find a constant
Obtain one of B = −4, C = 4 and D = 0
Obtain a second value
Obtain the third value

(ii) Integrate and obtain 3x – 4ln x
Integrate and obtain term of the form k ln(x2 +2)
Obtain term 2ln(x2 +2)
Substitute limits in an integral of the form axb ln x + c ln(x2 + 2) , where abc ≠ 0
Obtain given answer 3 − ln 4 after full and correct working

### Question

(i) By differentiating $$\frac{1}{cosx}$$ , show that if y = sec x then$$\frac{dy}{dx}$$ = sec x tan x.

(ii) Show that $$\frac{1}{(secx-tanx)}$$ ≡ sec x + tan x.

(iii) Deduce that $$\frac{1}{(secx-tanx)^{2}}$$ ≡ $$2 sec^{2}$$
x − 1 + 2 sec x tan x.

(iv) Hence show that $$\int_{0}^{\frac{1}{4}\pi }\frac{1}{(secx-tanx)^{2}}dx$$=$$\frac{1}{4}(8\sqrt{2}-\pi )$$

(i) Use correct quotient or chain rule Obtain the given answer correctly having shown sufficient working
(ii) Use a valid method, e.g. multiply numerator and denominator by sec x + tan x, and a version of Pythagoras to justify the given identity
(iii) Substitute, expand
(sec x +$$tan x)^{2}$$
and use Pythagoras once
Obtain given identity
(iv) Obtain integral 2 tan x – x + 2 sec x
Use correct limits correctly in an expression of the form a tan x + bx + c sec x, or
equivalent, where abc 0
Obtain the given answer correctly

### Question

By first expressing $$\frac{4x^{2}+5x+3}{2x^{2}+5x+2}$$ in partial fractions, show that

$$\int_{0}^{4}\frac{4x^{2}+5x+3}{2x^{2}+5x+2}$$dx =8-9In9

### Question

The diagram shows the curve y$$=8sin\frac{1}{2}x-tan\frac{1}{2}x$$ for 0 ≤ x < π. The x coordinate of the maximum point is α and the shaded region is enclosed by the curve and the lines x = α and y = 0.

(i) Show that$$α \frac{2}{3}\pi$$

(ii) Find the exact value of the area of the shaded region.

(i) Differentiate to obtain  $$4cos\frac{1}{2}x-\frac{1}{2}sec^{2}\frac{1}{2}x$$

Equate to zero and find value of $$cos\frac{1}{2}x$$

Obtain $$cos\frac{1}{2}x=\frac{1}{2}$$ and confirm $$\alpha =\frac{2}{3}\pi$$

(ii) Integrate to obtain $$-16cos\frac{1}{2}x$$

$$+2Incos\frac{1}{2}x$$or equivqlent

Using limits 0 and $$\frac{2}{3}\pi in acos\frac{1}{2}x+bIncos\frac{1}{2}x$$

Obtain$$8+2In\frac{1}{2}$$  or exact equivalent