Page 300 - Calculus Demystified

Chapter 4

3 3 287
2
2
2
(b) ln x dx = ln x + C
x 4

1
2
(c) sin x · cos xdx = sin x + C
2

1
2
(d) tan x · ln cos xdx =− ln cos x + C
2

2
(e) sec x · e tan x dx = e tan x + C
1
2
43
2
(f) (2x + 1) · (x + x + 7) dx = (x + x + 7) 44 + C
44
3. (a) We have
! "
2 j j 1
k 2
2
x + xdx = lim 1 + + 1 + ·
1 k→∞ k k k
j=1
k 2
2j j j 1
= lim 1 + + + 1 +
k→∞ k k 2 k k
j=1
k 2
2 3j j
= lim + +
k→∞ k k 2 k 3
j=1
2 3 2
2 k + k 3 2k + 3k + k 1
= lim k · + · + ·
k→∞ k 2 k 2 6 k 3

3 3 1 1 1
= lim 2 + + + + +
k→∞ 2 2k 3 2k 6k 2
3 1
= 2 + +
2 3
23
= .
6
(b) We have
2
2j
1 2 −1 +
k
x k 2
− dx = lim − ·
−1 3 k→∞ 3 k
j=1
k 2
−2 4j 4j
= lim 1 − +
k→∞ 3k k k 2
j=1