Page 217 - Calculus Workbook For Dummies

201
Chapter 11: Integration Rules for Calculus Connoisseurs

17. Integrate # dx . Hint: This is a 18. Last one: # 4 x - 1
2
625 x - 121 x dx. Same hint as in
2
2
2
u - a problem where u = seci. problem 17.
a
Solve It Solve It

Partaking of Partial Fractions

The basic idea behind the partial fractions technique is what I call “unaddition” of
1 1 2 2
fractions. Because + = , had you started with , you could have taken it apart —
2 6 3 3
or “unadded” it — and arrived at 1 + 1 . You do the same thing in this section except
2 6
that you do the unadding with rational functions instead of simple fractions.

Q. Integrate # 3 x dx 4. Plug the roots of the linear factors into
2
x - 3 x - 4 x one at a time.
A. 3 ln x + 1 + 12 ln x - 4 + C Plug in : 4 3 4 = B 4 + 1h B = 12
$
^
5 5 5
1. Factor the denominator. Plug in - : 1 - 3 = - 5 A A = 3
5
= # 3 x dx
^ x + 1 ^h x - 4h 5. Split up the integral and integrate.
2. Break up the fraction. # 3 xdx = # dx + 12 # dx
3
2
3 x = A + B x - 3 x - 4 5 x + 1 5 x - 4
^ x + 1 ^h x - 4h x + 1 x - 4 = 3 ln x + 1 + 12 ln x - 4 + C
5 5
3. Multiply both sides by the denominator
of the fraction on the left.
4 +
3 x = A x - h B x + 1h
^
^