IB Math SL 2 – Integration Worksheet
1.
Name_______________
The function f is such that f  (x) = 2x – 2.
When the graph of f is drawn, it has a minimum point at (3, –7).
i) Find f (x)
(a)
ii) Hence find f(x).
(6)
(b)
Find f (0), f (–1) and f (–1).
(3)
(c)
Hence sketch the graph of f labelling it with the information obtained in part (b).
(4)
(Note: It is not necessary to find the coordinates of the points where the graph cuts the x-axis.)
(Total 13 marks)
2.
Find
3.
Let f(t) =
4.
Let f(x) =
5.
6.
 sin (3x  7) dx;
(a)
1

t3 1

1 
 – 5  . Find


2t 3 

e
(b)
–4 x
dx .
 f (t ) dt.
(Total 3 marks)
x 3 . Find
(a)
f (x);
(a)
Find
(b)
Given that
 f ( x)dx.
(b)

(Total 6 marks)
dx
, giving your answer in terms of m.
2x  3
m
0

m
0
dx
= 1, calculate the value of m.
2x  3
The derivative of the function f is given by f (x) = e–2x +
(Total 6 marks)
1
, x < 1.
1 x
The graph of y = f(x) passes through the point (0, 4). Find an expression for f(x).
7.
(Total 4 marks)
(Total 6 marks)
3
Let f be a function such that  f ( x) dx  8 .
0
(a)
Deduce the value of
(i)
(b)

d
c

3
0
2 f ( x ) dx ;
(ii)
  f ( x)  2dx .
3
0
f ( x  2)dx  8 , write down the value of c and of d.
(Total 6 marks)
1
8.
The function f is given by f(x) = 2sin(5x – 3).
(a)
Find f "(x).
(b)
Write down
 f ( x)dx .
(Total 6 marks)
9.
The table below shows some values of two functions, f and g, and of their derivatives f  and g .
x
1
2
3
4
f(x)
5
4
–1
3
g(x)
1
–2
2
–5
f (x)
5
6
0
7
g (x)
–6
–4
–3
4
Calculate the following.
(a)
d (f(x) + g(x)), when x = 4;
dx
(b)
 g'( x)  6dx .
3
1
(Total 6 marks)
10. Find the exact value of the following:
(4 marks each)
3
(a)
  4 x  3 dx
1
4
(b)

3

  6 x  2  dx
1
1
(c)

 
6
8

 dx
5x  6 

4
(d)

1
  4sin 2 x  cos
0
2

 dx
x
(Total 16 marks)
Verify the above answers using your GDCs
2