15
Created by T. Madas Created by T. Madas INTEGRATION BY REVERSE CHAIN RULE
Created by T. Madas
Created by T. Madas
INTEGRATION
BY REVERSE CHAIN RULE
Created by T. Madas
Created by T. Madas
Question 1
Carry out each of the following integrations.
1. ( ) ( )4 5
2 233 1 1
10x x dx x C− = − +∫
2. ( ) ( )1 1
2 3 32 211 4 1 4
6x x dx x C
−
− = − − +∫
3. 3 44sin cos sinx x dx x C= +∫
4. 2 31sin cos cos
3x x dx x C= − +∫
5. 2
2
1010 7
7
xdx x C
x
= − +−∫
6. 2 2
6 e 3ex xx dx C= +∫
7. 4 2 51tan sec tan
5x x dx x C= +∫
8. 4 41sec tan sec
4x x dx x C= +∫
9. sin 2 sin 21e cos 2 e
2
x xx dx C= +∫
10. ( )2ln 1
ln2
xdx x C
x= +∫
11. ( )32
2 2cos sin 2 sin
3x x dx x C= +∫
12. 32
32
14 5
5
dx x x C
x x
= + +
+∫
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 2
Carry out each of the following integrations.
1. ( )3
4
4
1ln 2
2 4
xdx x C
x= + +
+∫
2. 2
3
3
1ln 4
4 3
xdx x C
x= − − +
−∫
3. 2
2
42ln 1
1
xdx x C
x= − +
−∫
4. 2
3
3
3ln 1
1
xdx x C
x= + +
+∫
5. 2
2
2
3e 3ln e 1
e 1 2
x
x
xdx C= − +
−∫
6. 24sec
4ln tan 4lnsec 4lnsintan
xdx x C x x C
x= + = + +∫
7. ( )2
2
1ln 9 1
9 1 18
xdx x C
x= + +
+∫
8. 2cosec
ln 1 cot1 cot
xdx x C
x= − + +
+∫
9. 2
2
42ln 10
10
xdx x C
x= − +
−∫
10. ( )ln 2 12
2 1 ln 2
xx
xdx C
+= +
+∫
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 3
Carry out each of the following integrations.
1. 2
2
1ln 9
9 2
xdx x C
x= − +
−∫
2. 2
2
105ln 9
9
xdx x C
x= − +
−∫
3. 2
2
3 3ln 4 2
4 2 4
xdx x C
x= − − +
−∫
4. 2
3
3
1ln 1
1 3
xdx x C
x= + +
+∫
5. 2
2
2 6ln 6 1
6 1
xdx x x C
x x
+= + + +
+ +∫
6. 3
3
3
4e 4ln 1 e
1 e 3
x
x
xdx C= − − +
−∫
7. ( )ln 3 13
3 1 ln 3
xx
xdx C
+= +
+∫
8. ( )22
2
ln 5 35
5 3 ln 25
xx
xdx C
+= +
+∫
9. 2
2
2 1ln 4 2
4 2 2
xdx x x C
x x
−= − − +
− −∫
10. sin cos
ln sin cossin cos
x xdx x x C
x x
−= − + +
+∫
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 4
Carry out each of the following integrations.
1. ( )
( )2
2
32
11
41
xdx x C
x
−
= − − +−∫
2. 2 21 1 1cos sin sin cos cos 2
2 2 4x x dx x C x C x C= + = − + = − +∫
3. 2
2
42 1 2
1 2
xdx x C
x
= − − +−∫
4. ( )2 2 31sec 1 tan tan tan
3x x dx x x C+ = + +∫
5. ( ) ( )22 1
sec 1 tan 1 tan2
x x dx x C+ = + +∫
6. ( )32
2sec tan sec 1 sec 1
3x x x dx x C+ = + +∫
7. 2 2 31tan sec tan
3x x dx x C= +∫
8. sin sine cos ex xx dx C= +∫
9. ( )322 2
sin cos sin3
x x dx x C= +∫
10. ( ) ( ) ( )2
2 2 4 3 21 1 32 1 1 1
2 2 2x x x dx x x C x x x x C+ + + = + + + = + + + +∫
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 5
Carry out each of the following integrations.
1. ( ) ( ) ( )2 22 1 sin 1 cos 1x x x dx x x C+ + + = − + + +∫
2. ( ) ( ) ( )2 211 cos 2 1 sin 2 1
2x x x dx x x C+ + + = + + +∫
3. ( ) ( )
3 2
1 1
1 ln 2 1 lndx C
x x x
= − ++ +∫
4. 4 514 cos sin 4 cos
5x x dx x x C− = + +∫
5. 2 2
3
cos 1 1cosec cot
sin 2 2
xdx x C x C
x= − + = − +∫
6. ( )32
2
1 2 tan 11 2 tan
cos 3
xdx x C
x
+= + +∫
7. cos
2 sinsin
xdx x C
x= +∫
8. 1
ln lnln
dx x Cx x
= +∫
9. 2 4 3
1 1
cos tan 3tandx C
x x x= − +∫
10. 3 41sin 2 cos 2 sin 2
8x x dx x C= +∫
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 6
Carry out each of the following integrations.
1. ( )
( )cos ln
sin lnx
dx x Cx
= +∫
2. 2
2
3 34 2
24 2
xdx x C
x
= − − +−∫
3. 3
4
sin 1sec
cos 3
xdx x C
x= +∫
4. 3 41cos sin sin
4x x dx x C= +∫
5. 2
3
4
sin 1tan
cos 3
xdx x C
x= +∫
6. ( ) ( )e sin e cos ex x xdx C= − +∫
7. 4 51sin 2 cos 2 cos 2
10x x dx x C= − +∫
8. ( ) ( )3 52 22 3 31
3 4 2 4 25
x x dx x C− = − − +∫
9. ( )2
23
3 2
1 32 3
42 3
xdx x x C
x x
+= + + +
+ +∫
10. 21 1sin 2 cos 2 sin 2 or cos 4
4 8x x dx x C x C= + − +∫
Created by T. Madas
Created by T. Madas
Question 7
Carry out each of the following integrations.
1. ( )
( )2
3ln 1ln
3
xdx x C
x= +∫
2. ( )( ) ( )4 5
2 211 2 1 2 1
10x x x dx x x C+ + + = + + +∫
3. 4 51sin cos cos
5x x dx x C= − +∫
4. 3 31sec tan sec
3x x dx x C= +∫
5. ( ) ( )4 5
2 213 3
10x x dx x C+ = + +∫
6. 3
cos 2
sinsin
xdx C
xx
= − +∫
7. 32cos sin sin
3x x dx x C= +∫
8. ( ) ( )
2
3 2
sec 1
1 tan 2 1 tan
xdx C
x x
= − ++ +∫
9. sin cos 1 2
cos 2 1 cos2 2cos 2 1
x xdx x C x
x= − + + = −
+∫
10. ( )2
2lnln
xdx x C
x= +∫
Created by T. Madas
Created by T. Madas
Question 8
Carry out each of the following integrations.
1. ( ) ( )3 52 22 3 31
3 4 2 4 25
x x dx x C− = − − +∫
2. e
2ex
xdx C
x= +∫
3.
31 22
1 41
3
xdx x C
x
+= + +
∫
4. 2
21
1
xdx x C
x
= + ++∫
5. 2
12 tan
cosdx x C
x x= +∫
6. 2
21
1
xdx x C
x
= + ++∫
7. 1
4 11
dx x C
x x
= + +
+∫
8. 4
5
sin 1sec
cos 4
xdx x C
x= +∫
9. ( )322 21
1 13
x x dx x C− = − − +∫
10.
31 22
2 3 22 3
4 3
xdx x C
x
+= + +
∫
Created by T. Madas
Created by T. Madas
Question 9
Carry out each of the following integrations.
1. ( )2
20
24 2 1
4
xdx
x
= −+∫
2. ( )
36
0
1ln16
2dx
x x
=+∫
3.
3
2
0
1ln 2
9 2
xdx
x=
+∫
