Download Integration (1)

Higher Mathematics
integration
[SQA]
1. Find
Z Part
[SQA]
2. Find
Z
3. Find
Part
Z
Level
C
3
Calc.
NC
Content
C12
Answer
U2 OC2
1989 P1 Q5
( x2 − 2)( x2 + 2)
dx , x 6= 0.
x2
Marks
4
•1
•2
•3
•4
2x + 3 dx .
Marks
3
Part
[SQA]
2
ss:
pd:
pd:
pd:
Level
C
Calc.
CN
4
Content
C14, C12, C13
start to write in standard form
complete process
integrate
integrate a −ve index
Answer
1 3
−1 + c
3 x + 4x
U2 OC2
2001 P2 Q6
x4 − 4
x2
2
2
• x − 4x−2
•3 13 x3 + c
−4x−1
•4
−1
•1
x2 − 5
√ dx .
x x
Marks
2
2
hsn.uk.net
Level
C
A/B
4
Calc.
NC
NC
Content
C14
C13
Page 1
Answer
U2 OC2
1999 P1 Q20
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
Higher Mathematics
[SQA]
4. Evaluate
Part
[SQA]
5. Find
Part
Z 2
1
Marks
5
Z
1
x +
x
2
2
Level
C
dx .
Calc.
NC
5
Content
C12, C13, C15
6. Find
Part
[SQA]
1998 P1 Q12
2
Marks
Level
Calc.
Content
2
A/B
CN
C22, C14
Z Answer
1
+c
3(7 − 3x)
U3 OC2
2000 P2 Q10
•1 −11 (7 − 3x)−1
•2 × −13
3x3 + 4x dx .
Marks
3
7. Evaluate
Part
U2 OC2
1
dx .
(7 − 3x )2
•1 pd: integrate function
•2 pd: deal with function of function
[SQA]
Answer
Z 0
−3
Marks
4
hsn.uk.net
Level
C
2x + 3
Level
C
3
Calc.
NC
2
Content
C12
Answer
U2 OC2
1994 P1 Q1
dx .
Calc.
NC
4
Content
C22, C15
Page 2
Answer
U3 OC2
1996 P1 Q5
c SQA
Questions marked ‘[SQA]’ c
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Higher Mathematics
[SQA]
8. Find the value of
Part
Marks
4
1
9. Evaluate
Z 4
A.
−2
B.
7
− 16
C.
1
2
D.
2
Key
D
1
1
Level
C
A/B
2u2
Calc.
NC
NC
du.
5
Content
C15
C15
Answer
U2 OC2
1989 P1 Q16
x −1/2 dx .
Outcome
2.2
hsn.uk.net
Z 2 2
u +2
2
Grade
C
Facility
0.71
Disc.
0.64
Calculator
NC
Page 3
Content
C13
Source
HSN 086
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
Higher Mathematics
[SQA]
10. For all points on the curve y = f ( x ) , f ′ ( x ) = 1 − 2x .
If the curve passes through the point (2, 1) , find the equation of the curve.
Part
[SQA]
Marks
3
Level
C
Calc.
NC
Content
C18
Answer
4
U2 OC2
1990 P1 Q8
11. A curve with equation y = f ( x ) passes through the point (2, −1) and is such that
f ′ ( x ) = 4x3 − 1.
Express f ( x ) in terms of x .
Part
[SQA]
Marks
5
Level
C
Calc.
NC
5
Content
C18
Answer
U2 OC2
1991 P1 Q10
dy
= 3x2 + 1 passes through the point (−1, 2) .
dx
Express y in terms of x .
12. A curve for which
Part
Marks
4
hsn.uk.net
Level
C
Calc.
NC
Content
C18
Page 4
Answer
4
U2 OC2
1992 P1 Q4
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
Higher Mathematics
[SQA]
13. A point moves in a straight line such that its acceleration a is given by
1
a = 2(4 − t) 2 , 0 ≤ t ≤ 4. If it starts at rest, find an expression for the velocity
v where a = dv
dt .
Part
Marks
4
Level
C
Calc.
NC
Content
C18, C22
•1 ss: know to integrate acceleration
•2 pd: integrate
•3 ic: use initial conditions with const.
of int.
4
• pd: process solution
[SQA]
Answer
3
V = − 43 (4 − t) 2 +
2
•3 0 = 2 ×
•4 c = 10 23
π
9,1
If f ′ ( x ) = sin(3x ) express y in terms of x .
•1
•2
•3
•4
[SQA]
Marks
4
ss:
pd:
ic:
pd:
Level
A/B
Calc.
NC
Content
C18, C23
U3 OC2
32
3
2002 P2 Q8
R
1
•1 V = (2(4 − t) 2 ) dt stated or implied
by •2
3
•2 2 × −13 (4 − t) 2
14. The graph of y = f ( x ) passes through the point
Part
4
3
1
( 4 − 0) 2
− 23
+c
.
Answer
y = − 13 cos(3x) +
4
U3 OC2
7
6
2000 P1 Q8
R
•1 y = sin(3x) dx stated or implied by
•2
•2 − 13 cos(3x)
•3 1 = − 31 cos( 3π
9 ) + c or equiv.
7
4
• c=6
know to integrate
integrate
interpret ( π9 , 1)
process
15.
Z
π
2
cos 2x dx .
3
(b) Draw a sketch and explain your answer.
2
(a) Evaluate
Part
(a)
(b)
(b)
Marks
3
1
1
hsn.uk.net
0
Level
A/B
C
A/B
Calc.
NC
NC
NC
Content
C23
T1, C16
T1, C16
Page 5
Answer
U3 OC2
1992 P1 Q14
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
Higher Mathematics
[SQA]
16. Differentiate sin3 x with respect to x .
Hence find
Part
17. Find
[SQA]
Z
sin2 x cos x dx .
Marks
1
3
Z
Level
C
A/B
Answer
U3 OC2
1994 P1 Q17
1
3
1
3 (2x
− 1) 2 + c
B.
1
2 (2x
− 1) − 2 + c
C.
1
2 (2x
− 1) 2 + c
D.
1
3 (2x
− 1) − 2 + c
Key
A
Outcome
3.2
Part
Content
C21, C19
C21, C19
(2x − 1) 2 dx where x > 12 .
A.
18. Find
Calc.
NC
NC
4
Z √
1
3
1
Grade
C
2
Facility
0
Disc.
0
Calculator
NC
Content
C22
1 + 3x dx and hence find the exact value of
Marks
4
hsn.uk.net
Level
A/B
Calc.
NC
Content
C22
Page 6
Answer
Z 1√
0
Source
2012 P1 Q14
1 + 3x dx .
4
U3 OC2
1993 P1 Q16
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
Higher Mathematics
19. Find
A.
R
− 25 x −5 − 5 sin 5x + c
B.
− 25 x −5 + 51 sin 5x + c
C.
− 23 x −3 + 51 sin 5x + c
D.
− 23 x −3 − 5 sin 5x + c
Key
C
[SQA]
(2x −4 + cos 5x ) dx .
Outcome
3.2
Grade
C
2
Facility
0
Disc.
0
Calculator
CN
Content
C13, C23
Source
2010 P1 Q9
√ dy
3 .
= 3 sin(2x ) passes through the point 5π
,
12
dx
Find y in terms of x .
20. A curve for which
Part
•1
•2
•3
•4
Marks
4
pd:
pd:
ss:
pd:
Level
A/B
Calc.
CN
Content
C18, C23
integrate trig function
integrate composite function
use given point to find “c”
evaluate “c”
hsn.uk.net
Page 7
Answer
√
y = − 32 cos(2x) + 41 3
4
U3 OC2
2001 P2 Q10
R
•1 3 sin(2x) dx stated or implied by •2
•2 − 32 cos(2x)
√
5
•3 3 = − 23 cos(2 × 12
π) + c
√
•4 c = 14 3 (≈ 0·4)
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
Higher Mathematics
[SQA]
21.
(a) Show that (cos x + sin x )2 = 1 + sin 2x .
(b) Hence find
Part
(a)
(b)
[SQA]
22. Find
Part
[SQA]
Marks
1
3
Z Z
1
(cos x + sin x )2 dx .
Level
C
A/B
Calc.
NC
NC
Content
T8
C23
3
Answer
U3 OC2
1993 P1 Q19
6x2 − x + cos x dx .
Marks
4
Level
C
Calc.
NC
4
Content
C23
Answer
U3 OC2
1995 P1 Q3
π
23. The curve y = f ( x ) passes through the point ( 12
, 1) and f ′ ( x ) = cos 2x .
Find f ( x ) .
Part
Marks
3
hsn.uk.net
3
Level
A/B
Calc.
NC
Content
C23
Page 8
Answer
U3 OC2
1997 P1 Q15
c SQA
Questions marked ‘[SQA]’ c
All others Higher Still Notes
Higher Mathematics
[SQA]
24.
(a) By writing sin 3x as sin(2x + x ) , show that sin 3x = 3 sin x − 4 sin3 x .
(b) Hence find
Part
(a)
(a)
(b)
Marks
2
2
4
Z
sin3 x dx .
Level
C
A/B
A/B
Calc.
NC
NC
NC
4
Content
T8, T8
T8, T8
C23
Answer
U3 OC2
1995 P2 Q9
[END OF QUESTIONS]
hsn.uk.net
4
Page 9
c SQA
Questions marked ‘[SQA]’ c Higher Still Notes
All others