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Title: Integration exercises
Description: integration exercises and answers to further develop integrating skills, this is my notes from university mechanical engineering course
Description: integration exercises and answers to further develop integrating skills, this is my notes from university mechanical engineering course
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FY009 INTEGRATION TUTORIAL 1
(1)
Estimate the area under the curve of the function y x 4 for 0 x 2 using 4
subintervals, working to four decimal places throughout
...
(2)
Evaluate the following integrals
...
Where appropriate expand the
brackets
...
25, 0
...
25 and 1
...
A 0
...
254 0
...
254 1
...
0703
...
4
5
5 0
0
2
4
(2)
(a)
x9
c
9
(3)
(a)
x 3
c
3
(c)
5x 9 / 5
c
9
x3
x 2 4x c
3
(b)
x7 x5
x3 2x c
7
5
(c)
1
c
2x 2
(d)
2x 3 / 2
2 x c
3
(e)
x 3 5x 2
6x c
3
2
(f) 2 x 3
(g)
x2 2
5
2 c
2 x 2x
(h) 3x 3 3x 2 x c
(b)
(d)
2
7
5x 2
4x c
2
(i) g ( x) x 4 2 x 3 x c
Final Answer 34
...
25
4
(k) g ( x)
2x 3 / 2
x4 x
3
Final Answer 5
...
2
5
Final Answer
78
15
...
5
2
INTEGRATION EXERCISES 2
1
...
(a)
sin(5x) dx
(c)
sin 3x cos 4 x 5e
(e)
4 ln( x) dx
(g)
1
16 x
2
(b)
dx
dx
(d)
3x x dx
(f)
2x
4 cos(8x) dx
e
(h)
x
1
(i)
x sin(x)dx
2
...
(You may assume
2
0
x
sin x cos 2 dx
0
3x
5
(j)
0
(k)
1
( x 1) 2
ln(x)
and y
81
ln(10)
ln( x) ( x 1) 2
for x in the given range)
...
3
...
cos(2 x) cos 2 x sin 2 x to show that sin 2 x
2
2
(b) Use the answer to part (a) to show sin 2 x dx
x sin(2 x)
c
...
cos 3x sin 4 x 5e 2 x
c (d)
3
4
2
4 x ln x 4 x c
(e)
(g)
(i)
(k)
10
2
...
5
0
ln( x)
(f)
(h)
(j)
(l)
sin(8 x)
c
2
3 2
x ln( x) c
2
e 3 x e 3 x
c
3
4
x
tan 1 c
3
3
0
...
3054
9
ln(10) dx 10 ln 10 6
...
1
( x 1) 2
x 2 2x 1
81 dx 81 81 81 dx 3
1
1
10
10
Hence the required area is 6
...
0913
...
(a) From the first of the trig equations we get cos 2 x 1 sin 2 x
...
Rearranging to make sin 2 x the subject gives sin 2 x
1 cos(2 x)
which is
2
2
the required result
...
(b) From part (a) we get sin 2 x dx
Title: Integration exercises
Description: integration exercises and answers to further develop integrating skills, this is my notes from university mechanical engineering course
Description: integration exercises and answers to further develop integrating skills, this is my notes from university mechanical engineering course