Math 126-104 Study Guide Fall 2014 Carter
General.
I have compiled about 4 or 5 problems of each type that I am considering to ask on the second test that will be held Monday, Nov. 3. If you can work these, and work similar problems, you will do well on the test. You have plenty of time to practice.
Compute the following integrals:
1.
(a)
Z x dx x 2 + 25
(b)
Z x dx x (ln( x )) 2
(c)
Z sec (3 x ) dx
(d)
Z e x dx
1 + e x
2.
(a)
(b)
Z e
√ x
√ x dx
Z
1
√
2 x 2 x
2 dx
(c)
Z log
3 x
( x ) dx
(d)
3.
(a)
Z
0
− 2
3
− x dx
Z
√
3 / 2 p
1 − x 2 dx
1 / 2
(b)
Z
√ x dx
1 − x 2
(c)
Z dx
√ x 2 + 9
(d)
Z x p x 2 − 4 dx

6.
(a)
(b)
(c)
(d)
7.
(a)
(b)
4.
(a)
(b)
(c)
(d)
5.
(a)
(b)
(c)
Z cos
3
( x ) sin
3
( x ) dx
Z cos (2 x ) sin
2
(2 x ) dx
Z x sec
2
( x
2
+ 1) dx
Z sin
3
( x ) dx
Z ln ( x ) dx
Z xe
2 x dx
Z arctan ( x ) dx
Z e ax sin ( bx ) dx
Z x − 5
( x + 2) 2 ( x − 1) dx
Z dx
( x − 1)( x − 5) dx
Z dx
( x 2 + 1)( x − 1)
Z dx
( x + 3)( x − 1) 2
Z
∞
0 dx x 2 + 1
Z
1
0 dx x 1 / 3

(c)
Z
∞
1 dx x 1001 / 1000
(d)
Z
∞ e
− x dx
0
8. Determine the limit of the sequence :
(a) a n
= ln n n
(b) a n
= n + 3 n 2 + 7 n + 6
(c) a n
= 2 + ( − 0 .
1) n
(d) a n
= n
2 n
(e)
8 a n
= 1 + n n
(f) a n
=
5 n n !
9. Give a formula for the n th partial sum of the series and use this formula to sum the series:
(a)
∞
X
2 k 2
2
−
( k + 1) 2 k =1
(b)
∞
X
1 k =1 k ( k + 1)
(c)
∞
X tan ( n ) − tan ( n − 1) k =1
10. Represent the repeating decimal as a fraction in lowest terms.
(a)
0 .
142857 = 0 .
142857142857142857 . . .
(b)
0 .
09 = 0 .
09090909 . . .

(c)
1 .
14 = 1 .
141414141414 . . .
11. Use any test that you like to determine if the given series converges (8 points each) .
(a)
∞
X
1 n =2 n (ln ( n )) 2
(b)
∞
X
( n + 1) n !
n =1
(c)
∞
X
1 n =1 p n ( n + 1)( n + 2)
(d)
∞
X
1 n
√ n + 5 n =2
(e)
∞
X
( n !)
2 n =1
(2 n )!