Calculus Volume 1

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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