James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 7.5 Strategy for Integration 507
The functions that we have been dealing with in this book are called elementary
functions. These are the polynomials, rational functions, power functions sx n d, exponential
func tions sb x d, logarithmic functions, trigonometric and inverse trigonometric functions,
hyperbolic and inverse hyperbolic functions, and all functions that can be obtained
from these by the five operations of addition, subtraction, multiplication, division, and
composition. For instance, the function
f sxd −Î x 2 2 1
x 3 1 2x 2 1
1 lnscosh xd 2 xe
sin 2x
is an elementary function.
If f is an elementary function, then f 9 is an elementary function but y f sxd dx need not
be an elementary function. Consider f sxd − e x 2 . Since f is continuous, its integral exists,
and if we define the function F by
Fsxd − y x
e t 2 dt
then we know from Part 1 of the Fundamental Theorem of Calculus that
0
F9sxd − e x 2
Thus f sxd − e x 2 has an antiderivative F, but it has been proved that F is not an elementary
function. This means that no matter how hard we try, we will never succeed in evaluating
y e x 2 dx in terms of the functions we know. (In Chapter 11, however, we will see how to
express y e x 2 dx as an infinite series.) The same can be said of the following integrals:
y e x
x dx y sinsx 2 d dx y cosse x d dx
y sx 3 1 1 dx
y
1
ln x dx y sin x dx
x
In fact, the majority of elementary functions don’t have elementary antiderivatives. You
may be assured, though, that the integrals in the following exercises are all elementary
functions.
1–82 Evaluate the integral.
cos x
1. y
1 2 sin x dx 2. y 1
s3x 1 1d s2 dx
0
3. y 4
sy ln y dy 4. y sin3 x
1 cos x dx
5. y
7. y 1 21
t
t 4 1 2 dt 6. y 1 x
0 s2x 1 1d dx 3
e arctan y
1 1 y 2 dy 8. y t sin t cos t dt
11. y
1
dx 12. y 2x 2 3
x 3 sx 2 2 1 x 3 1 3x dx
13. y sin 5 t cos 4 t dt 14. y lns1 1 x 2 d dx
15. y x sec x tan x dx 16. y s2y2
17. y
t cos 2 t dt 18. y 4
0
0
1
e st
st
19. ye x1ex dx 20. y e 2 dx
x 2
dx
s1 2 x 2
dt
9. y 4
2
x 1 2
x 2 1 3x 2 4 dx
10. y coss1yxd
x 3 dx
21. y arctan sx dx 22. y
ln x
xs1 1 sln xd 2 dx
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