Page 221 - Calculus Workbook For Dummies

205
Chapter 11: Integration Rules for Calculus Connoisseurs

Solutions for Integration Rules

a # x cos 5 - 2h dx = 1 x sin 5 - h 1 cos 5 - h C
x
2 +
x
2 +
x
^
^
^
5 25
1. Pick x as your u, because the algebraic function x is the first on the LIATE list.
2. Fill in your box.
x 1 sin(5x–2)
5
diff int

dx cos(5x–2)dx

2 - #
3. Use the “7” rule: # x cos 5 - 2h dx = 1 x sin 5 - h 1 sin 5 - 2h dx
x
x
x
^
^
^
5 5
1 1
x
2 +
4. Finish by integrating: = x sin 5 - h cos 5 - h C
x
2 +
^
^
5 25
b # arctan x dx = x arctanx - 1 ln 1 + x i + C
2
_
2
1. Pick arctan x as your u. You’ve got no choice.
2. Do the box thing.
arctan x x
diff int
dx dx
1 + x 2
3. Apply the “7” rule: # arctan x dx x arctan x - # xdx 2 = x arctan x - 1 ln 1 + x i + C
2
=
_
1 + x 2
π
c # x arctan x dx = 1 x 2 arctanx - 1 x + 1 arctanx + C
2 2 2
0
1. Pick arctan x as your u.
2. Do the box.
arctan x 1 x 2
2
diff int
dx xdx
1 + x 2