QUESTION ONE (COMPALSORY) - 30 MARKS

a) Work out the following integrals

i. ∫ 56 x

3dx

ii. ∫3 cos2 x dx

iii. ∫2

32

3 xdx

iv. ∫0

1

3 e3 t dt (10 Marks)

b) Given u=2 x3+3 x2 y+5 y3, find the following

i)∂ u2 x

ii)∂u∂ y

iii)∂u

∂ x2

iv) ∂2u∂ y2

v) ∂2ux∂ y

(6 Marks)

c) Find by integration method the area which is enclosed between the curve

y=2 x2−x+1, the x-axis and the ordinates x= -1, x= 2, given that the area is on the same

side of the x-axis.

(4 Marks)

d) The table shows the set of values of x and f(x), where f is a function of x.

x 1

.

0

0

1

.

2

5

1

.

5

0

1

.

7

5

2

.

0

0

2

.

2

5

2

.

5

0

2

.

7

5

3

.

0

0

f

(

x

)

2

.

0

0

0

1

.

7

9

0

1

.

6

3

3

1

.

5

1

2

1

.

4

1

4

1

.

3

3

3

1

.

2

6

5

1

.

2

0

6

1

.

1

5

5

Use the information and the numerical method stated with 8 intervals to estimate the value of

∫1

3

f (x )dx, correct to 3 decimal places

i. Simpson’s Rule

ii. Trapezoidal Rule

(5 Marks)

e) Evaluate∫R

∫ (6 x2¿+3 y2+2)d xd y ¿ , where

R: {0 ≤ x ≤ 1; 0 ≤ y ≤ 2}

(5 Marks)

QUESTION TWO-20 MARKS

a) Given that for any function f(x), the Taylor series expansion formula for f(a+h) is given

by:

f (a+b)=f (a )+h f I (a )+ h2 f II

2 !(a )+ h3 f III

3 !(a )+…,

i. Use the Taylor expansion series to write the first four terms of the expansion of

(2+h)5 ( 5 Marks)

ii. Hence, estimate the value of (2.03)5 and (1.8)5 correct to three decimal places.

(6 Marks)

b) Given u=cos x+sin y, show that

i.2ux2 +

2uy2 + u= 0

ii.2ux y

= 2uy x

QUESTION THREE- 20 MARKS

a) Evaluate ∫1

2

( 2x2 +

1x+ 3

4)dx ( 6 Marks)

b) Use the identity 2sinA cosB = sin (A+B) + sin (A-B) to express sin7tcos3t in the form

sin at+sin bt, where a and b are constants. Hence evaluate ∫0

1

sin 7 tcos 3 t dt

(4

Marks)

c) Using the substitution u=2 x2−3, express dx in terms of du and hence find the integral

∫2 x¿¿ ( 5 Marks)

d) Apply the integration formula for the basic function on the maths tables to determine the

following

i. ∫ 5

√4−t 2dt ( 2 Marks)

ii. ∫0

π2

3sin2θdθ (3 Marks)

QUESTION FOUR-20 MARKS

a) i) By resolving into partial fractions, show that

3 x−11(x−1)(x+3)

= 5x+3

− 2x−1 (7 Marks)

ii) Hence evaluate ∫2

33 x−11

(x−1)(x+3)dx correct to 3 decimal places (3 Marks)

b) i) Use integration by parts formula ∫udv=u v−∫ vdu with appropriate substitutions to

find the integral ∫ x ex dx (7 Marks)

ii) Hence evaluate ∫0

123

x ex dx (3 Marks)

QUESTION FIVE – 20 MARKS

a) The equation y=5 x2defines a curve. Find the area enclosed between the curve , the x-

axis and the ordinates x = 0 and x = 3 by

i) Integration

ii) Trapezoidal rule with 6 intervals

iii) Simpsons rule with 6 intervals ( 11 Marks)

b) Use integration method to find the volume of the solid of revolution formed when the

area in (a) above is revolved by one revolution about the x-axis. ( 4 Marks)

c) Use integration method to find the area of the region R which is enclosed by the curve

y=x2+1 and the straight line y=7−x ( 5 Marks)