5.5. OTHER OPTIONS FOR FINDING ALGEBRAIC ANTIDERIVATIVES
311
basic antiderivative.
(a)
x
2 sin(x
3 ) dx,
x
2 sin(x) dx,
sin(x
3 ) dx,
x
5 sin(x
3 ) dx
(b)
1
1 + x 2 dx,
x
1 + x 2 dx,
2x + 3
1 + x 2 dx,
e x
1 + (e x ) 2 dx,
(c)
x ln(x) dx,
ln(x)
x
dx,
ln(1 + x
2 ) dx,
x ln(1 + x
2 ) dx,
(d)
x
√
1 − x 2 dx,
1
√
1 − x 2
dx,
x
√
1 − x 2
dx,
1
x
√
1 − x 2
dx,
⊲⊳
The Method of Partial Fractions
The method of partial fractions is used to integrate rational functions, and essentially
involves reversing the process of finding a common denominator. For example, suppoes
we have the function R(x) =
5x
x 2 −x−2
and want to evaluate
5x
x 2 − x − 2
dx.
Thinking algebraically, if we factor the denominator, we can see how R might come from
the sum of two fractions of the form
A
x−2 +
B
x+1 . In particular, suppose that
5x
(x − 2)(x + 1)
=
A
x − 2
+
B
x + 1
.
Multiplying both sides of this last equation by (x − 2)(x + 1), we find that
5x = A(x + 1) + B(x − 2).
Since we want this equation to hold for every value of x, we can use insightful choices of
specific x-values to help us find A and B. Taking x = −1, we have
5(−1) = A(0) + B(−3),
and thus B =
5
3 . Choosing x = 2, it follows
5(2) = A(3) + B(0),
so A =
10
3 . Therefore, we now know that
5x
x 2 − x − 2
dx =
10/3
x − 2
+
5/3
x + 1
dx.