Solução_Calculo_Stewart_6e

F.
SECTION 7.6 TX.10 INTEGRATION USING TABLES AND COMPUTER ALGEBRA SYSTEMS ¤ 327
NowuseFormula22toget
x 2 2 2 + x 2 dx = x 8 (22 +2x 2 ) √ 2 2 + x 2 − 24
8 ln x + √ 2 2 + x 2 + C
= x 8 (2)(2 + x2 ) √ 4+x 2 − 2ln x + √ 4+x 2 + C
= 1 4 x(x2 +2) √ x 2 +4− 2ln √ x 2 +4+x + C
39. Maple gives x √ 1+2xdx= 1 10 (1 + 2x)5/2 − 1 6 (1 + 2x)3/2 , Mathematica gives √ 1+2x 2
5 x2 + 1
15 x − 1 15
,andDerive
gives 1 15 (1 + 2x)3/2 (3x − 1).Thefirst two expressions can be simplified to Derive’s result. If we use Formula 54, we get
x √ 1+2xdx= 2
15(2) 2 (3 · 2x − 2 · 1)(1 + 2x)3/2 + C = 1 30 (6x − 2)(1 + 2x)3/2 + C = 1
15 (3x − 1)(1 + 2x)3/2 .
41. Maple gives tan 5 xdx = 1 4 tan4 x − 1 2 tan2 x + 1 2 ln(1 + tan2 x), Mathematicagives
tan 5 xdx= 1 4 [−1 − 2cos(2x)] sec4 x − ln(cos x), and Derive gives tan 5 xdx= 1 4 tan4 x − 1 2 tan2 x − ln(cos x).
These expressions are equivalent, and none includes absolute value bars or a constant of integration. Note that Mathematica’s
and Derive’s expressions suggest that the integral is undefined where cos x 0 = {x | x 6= 0, |x| < 1} =(−1, 0) ∪ (0, 1). F has the same domain.
(b) Derive gives F (x) =ln √ 1 − x 2 − 1 − ln x and Mathematica gives F (x) =lnx − ln 1+ √ 1 − x 2 .
Both are correct if you take absolute values of the logarithm arguments, and both would then have the
same domain. Maple gives F (x) =− arctanh 1/ √ 1 − x 2 . This function has domain
x |x| < 1, −1 < 1/ √ 1 − x 2 < 1 = x |x| < 1, 1/ √ 1 − x 2 < 1 = x |x| < 1, √ 1 − x 2 > 1 = ∅,
the empty set! If we apply the command convert(%,ln); to Maple’s answer, we get
− 1
2 ln 1
√ +1 + 1
1 1 − x
2 2 ln 1
− √ , which has the same domain, ∅.
1 − x
2
45. Maple gives the antiderivative
x 2 − 1
F (x) =
x 4 + x 2 +1 dx = − 1 2 ln(x2 + x +1)+ 1 2 ln(x2 − x +1).
We can see that at 0, this antiderivative is 0. From the graphs, it appears that F has
amaximumatx = −1 and a minimum at x =1[since F 0 (x) =f(x) changes
sign at these x-values], and that F has inflection points at x ≈−1.7, x =0,and
x ≈ 1.7 [since f(x) has extrema at these x-values].