Name: __________________
Class:
Date: _____________
continuation
1 Evaluate the integral using the indicated trigonometric substitution.
Identify the associated right triangle.
1
x
2
x
1
4
2
dx ; x = 2sec 1
2
x
1
2
x
2
4
x
+ C
4
x
2
x
4 + C
2
c.
a.
1
4
x
2
x
4
2
x
4 + C
2
+ C
d.
b.
..to be continued
..to be continued
PAGE 1
Name: __________________
Class:
continuation
1
4
x
2
x
4
Date: _____________
2 Evaluate the integral using the indicated trigonometric substitution.
Identify the associated right triangle.
+ C
x
3
dx ; x = 6tan 2
x + 36
2
1 ( x + 108 )
3
x + 36 + C
( x 3 + 72)
x + 36 + C
e.
a.
1
3
2
b.
..to be continued
PAGE 2
Name: __________________
Class:
Date: _____________
continuation
1
3
( x2
72
)
continuation
2
1 ( x + 108 )
3
x + 36 + C
c.
e.
1
3
( x2
72
)
2
x + 36 + C
3 Evaluate the integral.
2 3
x
0
3
16 dx
x
2
4 Evaluate the integral.
d.
..to be continued
PAGE 3
7
2
x + 49 dx
0
2
x + 36 + C
Name: __________________
Class:
Date: _____________
5 Evaluate the integral.
8 Write out the form of the partial fraction decomposition of the
function. Do not determine the numerical values of the
coefficients.
dx
(x
a.
b.
c.
d.
e.
2
+ 2x + 2
1
2
tan
1
2
sec
1
2
tan
)2
1
( x + 1) +
(a)
x 2
2
+ C
(b)
x + 2x + 2
1
( x + 1) +
x + 2
2
1
( x + 1) +
x + 1
2
+ C
(a)
+ C
sec
1
2
tan
1
( x + 1) x + 1
1
(a)
2
( x + 1) x + 2x + 2
x + 1
+ C
Find the area of the region bounded by the hyperbola 9x
2
2
4y = 36 and the line x = 9 .
7 A water storage tank has the shape of a cylinder with diameter 10
ft. It is mounted so that the circular cross sections are vertical. If
the depth of the water is 7 ft, what percentage of the total capacity
is being used?
2
A
B
+
x + 9
3x + 4
A + Bx + C
2
x
(x + 9)
(a)
A
B
+
x + 9
3x + 4
(b)
A +
B
+
x
x + 9
c.
Please round your answer to nearest tenth.
________ %
3
b.
(b)
6
2
Ax + Bx + C x
x + 18 x + 81x
+ C
2
Ax + B + Cx + D
x + 9
3x + 4
3
(b)
x + 2x + 2
2
a.
x + 2x + 2
1
2
1
3
x + 18 x + 81x
x + 2x + 2
8x
(x + 9)(3x + 4)
C
(x + 9)
(a)
Ax + B + Cx + D
x + 9
3x + 4
(b)
A + Bx + Cx + D
2
x
x + 18x + 81
(a)
A + B + C + D
x
9
3x
4
d.
2
e.
(b) A +
x
PAGE 4
3
B
18 x
+
2
C
81x
2
Name: __________________
Class:
9 Write out the form of the partial fraction decomposition of the
function. Do not determine the numerical values of the
coefficients.
x
4
1
1
4
0
B +
(b) A +
6
2
A
x Ct + D
(t
2
+ 3
)
2
b.
B
+
+ Cx + D
2
1
x + 1
x + 1
At + B +
(b)
2
t + 6
Ct + D
c.
( t 2 + 3) 2
B
1
3
4 ln x + 4 + 2 ln x 2 3
3
4 tan 1
1 (2x 4)
2
4 2
2
x + 4 ln x + 4 + 2 ln x 3
3
4 tan 1
1 (2x 4)
3
4 3
2 ln x + 4 + 4 ln x 2 3
3
4 tan 1
1 (2x 4)
2
4 2
4 ln x + 4 + 2 ln x 2 3
3
4 tan 1
1 (2x 4)
3
4 3
x d.
c.
B +
(b) A +
t
6
2
(a) 1 +
A
x Ct + D
e.
B
+
+ Cx + D
2
1
x + 1
x + 1
(a)
At + B + Ct + D +
2
2
t + 6
t + 3
A
x 1
+
(b)
At + B + Ct + D
2
2
t + 6
t + 3
10 Evaluate the integral.
x 12
dx
(x + 4)(x 4)
PAGE 5
Et + F
( t 2 + 3) 2
B
+ Cx + D
2
x + 1
x + 1
e.
2 ln x + 4 + 4 ln x 2 3
3
2 tan 1 1 (2x 4)
3
8
x ( t 2 + 3)
d.
(b)
6
x b.
x
x x a.
4
61 dx
3
1
4
(a) A
54x x + 64
a.
(a)
x
2
x
B
(a) A +
t
3
2
( t 2 + 6 ) ( t 2 + 3) 2
x
9x
12 Evaluate the integral.
t + t + 3
(b)
11 Evaluate the integral.
4
x
(a)
Date: _____________
4x + 16
4x + 16
4x + 16
+
+
4x + 16
4x + 16
+
Name: __________________
Class:
Date: _____________
13 Evaluate the integral.
x (x
2
14 Make a substitution to express the integrand as a rational
function and then evaluate the integral.
3
+ 8x + 18
dx
)
x
2
3
3
dx
2
x + 7
7 2
tan
8
1
a.
2 (x
2
+ 8x + 18 )
7(x + 4)
(
2
4 x + 8x + 18
c.
(
2
4 x + 8x + 18
C
x + 4
)
2
)
+
x + 8x + 18
2
2 ( x + 8x + 18 )
7(x + 4)
(
2
4 x + 8x + 18
e.
(
2
2 x + 8x + 18
7(x + 4)
)
2
4 ( x + 8x + 18 )
x + 4
2
1
1
x + 4
2
x + 4
2
7 2
tan
8
1
+ C
(
2
5
3
(
tan
+ C
1
2
( x + 4) )
(
2
21 x + 7
4
(
2
+ 3 x + 7
4
(
2
+ 3 x + 7
2
5
3
)
3 x + 7
10
e.
5
3
)
)
3 x + 7
5
d.
5
3
(
2
)
3 x + 7
2
(
)
2
3
)
2
3
)
2
+ 21 x + 7
4
2
3
+ C
+ C
+ C
2
3
)
+ C
2
3
+ C
15 Make a substitution to express the integrand as a rational
function and then evaluate the integral.
e
e
a.
b.
c.
d.
e.
PAGE 6
(
2
3 x + 7
5
c.
x + 4
2
(
2
5
3
)
3 x + 7
10
b.
+
(
2
3 x + 7
10
a.
+ C
)
1
7 2
tan
8
7 2
tan
8
1
d.
1
+ C
7(x + 4)
b.
2x
2x
x
dx
+ 3e + 2
( e x + 2) ( e x + 1) + C
2
x
x
ln ( e + 2 ) ( e + 1 ) + C
2
x
(
)
e
+
2
ln
+ C
( e x + 1)
( e x + 2) + C
ln
( e x + 1)
2x
x
ln ( e + 3e + 2 ) + C
ln
Name: __________________
Class:
16 Evaluate the integral.
sin 3 x + sec 3 x dx
tan 3 x
a.
1 sin 3 x + 1 ln ( csc 3 x + cot 3 x ) + C
3
3
b.
1 cos 3 x + 1 ln ( sec 3 x 3
3
cot 3 x ) + C
c.
1 cos 3 x + 1 ln ( csc 3 x 3
3
cot 3 x ) + C
d.
1 sin 3 x 3
cot 3 x ) + C
e.
1 sin 3 x + 1 ln ( csc 3 x 3
3
1 ln ( csc 3 x 3
17 Evaluate the integral.
3
3
e
3
arctan y
dy
1 + y
2
3
18 Evaluate the integral.
1
(1
+
x
)
9
dx
0
19 Evaluate the integral.
1
dx
x 5x + 1
20 Evaluate the integral.
3
ln ( tan x ) dx
sin x cos x
4
PAGE 7
cot 3 x ) + C
Date: _____________