7/30/2019 Questions Section 5, Integration

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7/30/2019 Questions Section 5, Integration

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Tutorial 4: Integration

ECO4112F 2011

1. (x + 1)20 dx

2. x x 2 + 5 dx3. 4x 2 x 4 + 1

2dx

4. 2xe x2

dx

5. x2 + 1 ex

3 +3 x dx

6.

7x

dx

7. 2x3 + 3 x

x 4 + 3 x 2 + 7dx

8. 2x 1dx9. ex/ 4 dx

10. x2 + 1

2dx

11. e x

x dx

12. 1

x 1+

1(x 1)

2 dx

13. sin(5 x 3) dx14. 73 2x dx15. 2

3x dx

16. 9x2 + 53x

dx

17. 3ln x

x dx

18. x + 3x + 6 dx

19.3

1 3x2

x + 6 dx

20.1

0 e3 t dt

21.1

0 2x2 x 3 1

3dx

22.1

13 x 5 dx

23. 103 x dx24. ln x x dx25.

2

1 x ln xdx26. xe x

2

dx

27.

x 2 e2 x+1 dx

28. xe x dx29. y3 ln ydy30. x x + 1 dx31.

2

1 xe 2 x dx32. (2x 1)ln( x 1) dx33. ex sin xdx34.

e

1 x ln x2 dx

1

7/30/2019 Questions Section 5, Integration

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Tutorial 4: Integration

SELECTED SOLUTIONS

ECO4112F 2011

1. R (x + 1)20 dx

Easy

2. R xp x 2 + 5 dx=

x 2 + 53 =2

3+ C

3.

R 4x 2 x 4 + 1

2dx

= 4 x11

11+ 2x

7

7+ x

3

3+ C

4. R 2xe x2

dx

= ex2

+ C

5. R x2 + 1 ex

3 +3 x dx

=13

ex3 +3 x + C

6. R 7x

dx

Easy

7. Z 2x3 + 3 x

x 4 + 3 x 2 + 7dx

Easy

8. R p 2x 1dx=

13

(2x 1)3 =2 + C

9. R e x=4 dx

...

10. R x2 + 1 2 dx

=x 5

5+

2x 3

3+ x + C

11. Z ep x

p x dx

= 2 ep x + C

12. Z 1x 1 + 1(x 1)2 dx...

13.

R sin(5x 3) dx

= cos(5x 3)5

14. Z 73 2x dxSimple, but be careful of signs

15. R 23 x dx

=1

ln 223 x + C

16.

Z 9x 2 + 5

3x

dx

Hint: divide through

17. Z 3ln x

xdx

=3ln x

ln 3+ C

18. Z x + 3x + 6 dxHint: rewrite the top and break up intotwo fractions ...

19.3

R 1 3x2 x + 6 dx

= 48

1

7/30/2019 Questions Section 5, Integration

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20.1

R 0 e3 t dt

=1

3e3 1

21.1

R 0 2x2 x 3 1

3dx

=16

22.1

R 13p x 5 dx

= 0

23.

R p 103 x dx

= 2p 103 x3ln10

+ C

24. Z ln xp x dx= 2 p x (ln x 2) + C

25.2

R 1 x ln xdx= 2ln 2

34

26. R xe x2

dx

=12

ex2

+ C

27. R x2 e2 x+1 dx

=e2 x+1

2x 2 x + 12 + C

28. R xe x dxEasy, use by parts29. R y3 ln ydy

=y4

4ln y

14

+ C

30. R xp x + 1 dx=

215

(x + 1) 3 =2 (3x 2) + C

31.2

R 1 xe 2 x dx=

14

e2 3e2 1

32. R (2x 1)ln( x 1) dx= x (x 1)ln( x 1)

x 2

2+ C

33. R ex sin xdxDont use by parts more than twice34.

e

R 1 p x ln x 2 dxUse by parts.

2

7/30/2019 Questions Section 5, Integration

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E C O 4 1 1 2 F , T u t o r i a l 4

1 2

1x 1

+1

(1 x )

2dx = 1x

1dx + 1(x 1)2

dx

= ln ( x 1) + (x

1) 1

+ C

2 4

ln x x dx = (ln x ) x 12 dx

= 2 x12 ln x + A

2x

12

1x

dx

= 2 x12 ln x + A 2x

12 dx

= 2 x12 ln x + A 4

x12

B

= 2 x (ln x 2) +C

2 5

2

1

x ln xdx =12

x 2 ln x2

1 2

1

12

x 21x

dx

= 2ln2 12

ln 1 14

x 22

1

= 2 ln 2 34

3 4

e

1

x ln x 2 dx = e

12x

12 ln xdx

=43

x32 ln x

e

1 e

1

43

x32

1x

dx

=43

e32

89

x32

e

1

= 43e

32

89 e32 + 89

=49

e32 +

89

1