602
Chapter 5 | Integration
Solution First factor the 3 outside the integral symbol. Then use the u −1 rule. Thus,
⌠ ⌡
dx =3 ⌠ ⌡
3 x −10
1 x −10
dx
=3 ∫ du u =3ln| u | + C =3ln| x −10| + C , x ≠10.
See Figure 5.39 .
Figure 5.39 The domain of this function is x ≠10.
Find the antiderivative of 1
5.38
x +2 .
Example 5.46 Finding an Antiderivative of a Rational Function
3 +3 x x 4 +3 x 2 .
Find the antiderivative of 2 x
Solution This can be rewritten as ⌠ ⌡ ⎛
−1 dx . Use substitution. Let u = x 4 +3 x 2 , then
⎝ 2 x 3 +3 x ⎞ ⎠ ⎛
⎝ x 4 +3 x 2 ⎞ ⎠
du =4 x 3 +6 x . Alter du by factoring out the 2. Thus, du = ⎛
⎝ 4 x 3 +6 x ⎞
⎠ dx
⎛ ⎝ 2 x 3 +3 x ⎞
= 2
⎠ dx
⎛ ⎝ 2 x 3 +3 x ⎞
1 2
du =
⎠ dx .
Rewrite the integrand in u :
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