Sec7.3$ $ MATH$1152Q$ HW#(ANSWER#KEY) · Sec7.3$ $ MATH$1152Q$ $ HW#(ANSWER#KEY)$ $ Page 3 of 6...

Sec 7.3 MATH 1152Q HW (ANSWER KEY)

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1. Evaluate .

[Solution]

Let , where , then .

Thus

2. Evaluate .

[Solution]


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 =


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Thus

4. Evaluate .

[Solution] Let , then .

When , and when , .

Thus


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= +


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5. Let be the region bounded by the function and -axis on the

interval .

Find the area of the region . [Solution] 1

Thus


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=


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When , and when , .

Thus

6. Evaluate .

[Solution]


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= + + +


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7. Evaluate . [Solution]

Let where , then .

When , and when , .

Thus


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-

= +


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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

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