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Sec 7.3 MATH 1152Q HW (ANSWER KEY)
Page 1 of 6
1. Evaluate .
[Solution]
Let , where , then .
Thus
2. Evaluate .
[Solution]
Let , where , then .
Thus
3. Evaluate .
[Solution]
Let , where or , then .
When , and when , .
2
1 4
dxx x-ò
2sinx q=2 2p pq- £ £ 2cos dx dq q=
2
1 4
dxx x-ò 2
1 2cos 2sin 4 4sin
dq qq q
=-
ò1 1 2 sin
dqq
= ò1 csc 2
dq q= ò1 ln csc cot2
Cq q= - + +
21 2 4ln2
x Cx
+ -= - +
2
1 16dx
x +ò
4 tanx q=2 2p pq- < < 24sec dx dq q=
2
1 16dx
x +ò 2
2
1 4sec 16 tan 16
dq qq
=+
òsec dq q= òln sec tan Cq q= + +
2 16ln4
x x C+ += +
2
3 22
1 1dx
x x -ò
secx q= 02pq£ <
32pp q£ < sec tan dx dq q q=
2x =4pq = 2x =
3pq =
Sec 7.3 MATH 1152Q HW (ANSWER KEY)
Page 2 of 6
Thus
4. Evaluate .
[Solution] Let , then .
When , and when , .
Thus
Let , where , then .
When , and when , .
Thus
2
3 22
1 1dx
x x -ò 3
3 24
1 sec tan sec sec 1
dp
p q q qq q
=-
ò3
24
1 sec
dp
p qq
= ò23
4
cos dp
p q q= ò3
4
1 cos 2 2
dp
pq q+
= ò3
4
1 1 sin 22 4
p
pq qæ ö= +ç ÷
è ø
1 2 1sin sin6 4 3 8 4 2p p p pæ ö æ ö= + - +ç ÷ ç ÷è ø è ø
3 124 8 4p
= + -
220
cos 1 sin
x dxx
p
+ò
sinu x= cos du x dx=
0x = 0u =2
x p= 1u =
220
cos 1 sin
x dxx
p
+ò
1
20
1 1
duu
=+
ò
tanu q=2 2p pq- < < 2sec du dq q=
0u = 0q = 1u =4pq =
220
cos 1 sin
x dxx
p
+ò
1
20
1 1
duu
=+
ò24
20
1 sec 1 tan
dp
q qq
=+
ò
40
sec dp
q q= ò( ) 40ln sec tan
p
q q= +
( )ln 2 1= +
Sec 7.3 MATH 1152Q HW (ANSWER KEY)
Page 3 of 6
5. Let be the region bounded by the function and -axis on the
interval .
Find the area of the region . [Solution] 1
Thus
Let , where , then .
When , and when , .
Thus
[Solution] 2
Let , then .
R ( )2
12 3
f xx x
=- + +
x
[ ]0,1
R
A1
20
1 2 3
dxx x
=- + +
ò2 2 3x x- + + ( )2 2 3x x= - - +
( )2 2 1 1 3x x= - - + + +
( )21 4x= - - +
A1
20
1 2 3
dxx x
=- + +
ò
( )1
20
1 4 1
dxx
=- -
ò
1 2sinx q- =2 2p pq- £ £ 2cos dx dq q=
0x =6pq = - 1x = 0q =
A( )
1
20
1 4 1
dxx
=- -
ò0
26
1 2cos 4 4sin
dp q qq-
=-
ò0
6
1 dp q-
= ò
6p
=
A( )
1
20
1 4 1
dxx
=- -
ò1
20
1 12 1
2
dxx
=-æ ö- ç ÷
è ø
ò
12xu -
=1 2
du dx=
Sec 7.3 MATH 1152Q HW (ANSWER KEY)
Page 4 of 6
When , and when , .
Thus
6. Evaluate .
[Solution]
Let , where , then .
Thus
0x = 12
u = - 1x = 0u =
A1
20
1 12 1
2
dxx
=-æ ö- ç ÷
è ø
ò
01 22
1 1
duu-
=-
ò
( )0112
sin u-
-=
6p
=
2
1 1 16
dxx+
ò
1 tan4
x q=2 2p pq- < < 21 sec
4dx dq q=
2
1 1 16
dxx+
ò 2
2
1 1 sec 41 tan
dq qq
=+
ò1 sec 4
dq q= ò1 ln sec tan4
Cq q= + +
21 ln 1 16 44
x x C= + + +
Sec 7.3 MATH 1152Q HW (ANSWER KEY)
Page 5 of 6
7. Evaluate . [Solution]
Let where , then .
When , and when , .
Thus
8. Evaluate . [Solution]
Let where or , then .
Thus
22
20
4x dxx-ò
2sinx q=2 2p pq- £ £ 2cos dx dq q=
0x = 0q = 2x =4pq =
22
20
4x dxx-ò
24
20
4sin 2cos 4 4sin
dp q q q
q=
-ò
240
4sin dp
q q= ò4
0
1 cos 24 2
dp q q-
= ò( ) 402 sin 2
p
q q= -
12p
= -
2 2
1 9 1
dxx x -ò
1 sec3
x q= 02pq£ <
32pp q£ <
1 sec tan 3
dx dq q q=
2 2
1 9 1
dxx x -ò
2 2
1 1 sec tan 1 3sec sec 19
dq q qq q
=-
ò
13 sec
dqq
= ò3 cos dq q= ò3sin Cq= +
29 1x Cx-
= +
Sec 7.3 MATH 1152Q HW (ANSWER KEY)
Page 6 of 6
9. Evaluate . [Solution]
Let , where , then .
Thus
25 4 x x dx+ -ò
25 4x x+ - ( )2 4 5x x= - - +
( )2 4 4 4 5x x= - - + + +
( )22 9x= - - +
2 3sinx q- =2 2p pq- £ £ 3cos dx dq q=
25 4 x x dx+ -ò ( )29 2 x dx= - -ò23cos 9 9sin dq q q= -ò
29 cos dq q= ò1 cos 29
2dq q+
= ò9 9 sin 22 4
Cq q= + +
9 9 sin cos2 2
Cq q q= + +
( )21 9 29 2 9 2sin2 3 2 3 3
xx x C- - -- -æ ö= + × × +ç ÷è ø
( )219 2 2sin 9 22 3 2
x x x C- - -æ ö= + - - +ç ÷è ø