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Created by T. Madas
Created by T. Madas
INTEGRATION
BY PARTS
Created by T. Madas
Created by T. Madas
Question 1
Carry out the following integrations:
1. 2 2 21 1e e e
2 4
x x xx dx x C= − +
2. 3 3
3 cos2 sin 2 cos 22 4
x x dx x x x C= + +
3. 1 1
sin 4 cos 4 sin 44 16
x x dx x x x C= − + +
4. 2 2
2 sin5 cos5 sin55 25
x x dx x x x C− = − +
5. ( ) ( )1 2 e 2 1 e 2ex x xx dx x C
− − −− = − + +
6. 2 3 2 3 3 31 2 2e e e e
3 9 27
x x x xx dx x x C
− − − −= − − − +
7. 3 4 416 ln 4 lnx x dx x x x C= − +
8. ln lnx dx x x x C= − +
9. ( ) ( ) ( )1 1 12 2 2
cos 2 sin 4cosx x dx x x x C= + +
10. ( ) ( ) ( ) ( ) ( )1 1
3 1 sin 3 1 3 1 cos 3 1 sin 3 13 3
x x dx x x x C− − = − − − + − +
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 2
Carry out the following integrations:
1. 3 3 326 e 2 e e
3
x x xx dx x C= − +
2. 4
12 cos3 4 sin3 cos33
x x dx x x x C= + +
3. 1 1
sin 6 cos6 sin 66 36
x x dx x x x C= − + +
4. 1 1
sin 2 cos2 sin 22 4
x x dx x x x C− = − +
5. ( ) ( )3 3 31 12 e 2 e e
3 9
x x xx dx x C
− − −− = − − + +
6. 2 4 2 4 4 41 1 1e e e e
4 8 32
x x x xx dx x x C= − + +
7. 1 1 1 1
2 22 2 2 2e 2 e 8 e 16ex x x x
x dx x x C− − − −
= − − − +
8. 4 5 525 ln 5 lnx x dx x x x C= − +
9. ( ) ( ) ( )2 2 23 3 3
24 cos 36 sin 54cosx x dx x x x C= + +
10. ( ) ( ) ( ) ( )2 2sin 1 cos 1 2 sin 1 2cos 1x x dx x x x x x C− = − + − − − +
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 3
Carry out the following integrations:
1. 4 4 412
1 1e e e
8 32
x x xx dx x C= − +
2. 5 5
5 sin 4 cos 4 sin 44 16
x x dx x x x C= − + +
3. ( ) ( )1 1
2 1 cos 2 2 1 sin 2 cos 22 2
x x dx x x x C+ = + + +
4. 3 3
3 cos4 sin 4 cos44 16
x x dx x x x C− = − − +
5. 2 2 2 2 2 21 1 1e e e e
2 2 4
x x x xx dx x x C
− − − −= − − − +
6. 2 21 2 2sin 5 cos5 sin 5 cos5
5 25 125x x dx x x x x x C= − + + +
7. 2 21 1 1 13 3 3 3
cos 3 sin 18 cos 54sinx x dx x x x x x C= + − +
8. 3 4 412
1 1ln ln
8 32x x dx x x x C= − +
9. 2 21 1ln 3 ln3
2 4x x dx x x x C= − +
10. 3 2 2
ln ln 1
2 4
x xdx C
x x x= − − +
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 4
Carry out the following integrations:
1. 5 5 51 1e e e
5 25
x x xx dx x C= − +
2. 2 2
2 cos3 sin 3 cos33 9
x x dx x x x C= + +
3. 1 1
sin 3 cos3 sin 33 9
x x dx x x x C= − + +
4. 1 1
sin 4 sin 4 cos 416 4
x x dx x x x C= − +
5. 2ln 2 ln 2x dx x x x C= − +
6. 2 3 31 1ln ln
3 9x x dx x x x C= − +
7. ( ) ( ) ( )1 1 12 2 2
sin 4sin 2 cosx x dx x x x C= − +
8. ( ) ( ) ( )1 1
sin 2 1 cos 2 1 sin 2 12 4
x x dx x x x C− = − − + − +
9. 2
ln ln 1x xdx C
x x x= − − +
10. 2sec tan ln secx x dx x x x C= − +
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 5
Carry out the following integrations:
1. 2 2sin cos 2 sin 2cosx x dx x x x x x C= − + + +
2. 3 4 41 1ln ln
4 16x x dx x x x C= − +
3. ( ) ( )sin ln sec cos 1 ln secx x dx x x C= − + +
4. 1 1
cos5 sin 5 cos55 25
x x dx x x x C= + +
5. 2 21 2 2sin 3 cos3 sin 3 cos3
3 9 27x x dx x x x x x C= − + + +
6. ( )2 23 34 e 3 2 3 e
x x
x dx x C− −
= − + +
7. 2 21 1 1 1cos 3 sin 18 cos 54sin
3 3 3 3x x dx x x x x x C
= + − +
8. 2 2 2 22 sec tan sec 2 tan 2ln secx x x dx x x x x x C= − + +
9. ( )1 12 22 2e 2 8 16 e
x x
x dx x x C= − + +
10. sec tan sec ln sec tanx x x dx x x x x C= − + +
Created by T. Madas
Created by T. Madas
Question 6
Carry out the following integrations:
1. ( )1 14 42 2e 4e 8 32
x x
x dx x x C− −
= − + + +
2. ( )2 2e e 2 2x xx dx x x C
− −= − + + +
3. ( )1
e cos e cos sin2
x xx dx x x C= + +
4. ( ) ( )2 2
ln ln 2 ln 2x dx x x x x x C= − + +
5. ( )1
e sin e sin cos2
x xx dx x x C= − +
6. ( ) ( )3 2 2 2 3 215 2 e e 4 14 14 1
8
x xx x dx x x x C+ − = + − − +
7. 2 21 1 1cos sin 2 cos 2
8 4 8x x dx x x x x C= + + +
8. ( )3 23ln 2 ln 2 2ln 1
4x x dx x x C= − +
Created by T. Madas
Created by T. Madas
Question 7
Carry out the following integrations, to the answer given:
1.
ln 2
2
0
3e ln 4
4
xx dx = −
2. 3
0
26 sin 3
3x x dx
π
π=
3. ( )2
2 2
0
1cos 8
4x x dx
π
π= −
4. ( )e
2
1
1ln e 1
4x x dx = +
5. ( )1
3 3
0
44 e 2e 1
9
xx dx = +
6. 4
0
sin 416
x x dx
π
π=
7.
2
3
1
15ln 4ln 2
16x x dx = −
8. ( )1
2 2
0
1e 1 3e
4
xx dx
− −= −
9. ( )4
0
312 cos 2 2
2x x dx
π
π= −
10. 2
4
4 sin 2 1x x dx
π
π
π= −
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 8
Carry out the following integrations, to the answer given:
1. 3
0
sin 39
x x dx
π
π=
2. 4
0
12 cos 4
4x x dx
π
= −
3.
ln 2
0
4 e 2 ln 4xx dx
−= −
4.
e
1
ln 1x dx =
5. 2
0
sin 24
x x dx
π
π=
6. 12
ln 4
0
e 8ln 2 4x
x dx = −
7. ( )0
1cos 2 2 4 16
4x x dx
π
π
= + −
8. ( )1
2 2
0
2 1 e exx dx+ =
9.
1
21e
1 3ln 1
4 ex x dx
= −
10. ( ) ( )0
1
32
3ln 2 3 ln 27 2x dx
−
+ = − REQUIRES ADDITIONAL TECHNIQUES
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 9
Carry out the following integrations, to the answer given:
1. ( )4
2
0
1sec ln 4
4x x dx
π
π= −
2. ( )2
2
1
ln 1ln 2
2
xdx
x=
3. ( )2 2
0
2 1sin 4
16x x dx
π
π= +
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