Evaluate using the tabular method shown in the notes.
integral 5x4 ex dx
__________ +C
Transcribed Image Text: Integration by Tables
So far in this chapter, you have studied several integration techniques that can be used
with the basic integration rules. But merely knowing how to use the various techniques
is not enough. You also need to know when to use them. Integration is first and
foremost a problem of recognition. That is, you must recognize which rule or technique
to apply to obtain an antiderivative. Frequently, a slight alteration of an integrand will
require a different integration technique (or produce a function whose antiderivative is
not an elementary function), as shown below.
Salm
x2
In x
2
x In x dx
+ C
Integration by parts
4
In x
dx
(In x)?
+ C
2
Power Rule
1
dx = In|In x| + C
In x
Log Rule
dx = ?
In x
Not an elementary function
Many people find tables of integrals to be a valuable supplement to the integration
techniques discussed in this chapter. Tables of common integrals can be found in
Appendix B. Integration by tables is not a "cure-all" for all of the difficulties that can
accompany integration-using tables of integrals requires considerable thought and
insight and often involves substitution.
Each integration formula in Appendix B can be developed using one or more of
the techniques in this chapter. You should try to verify several of the formulas. For
instance, Formula 4
1
du
(а + bu)?
и
а
+ Inla + bu| ) + C
Formula 4
b2
(а + bu
can be verified using the method of partial fractions, Formula 19
Ja + bu
du
du
= 2Ja + bu + a
Formula 19
uJa + bu
и
can be verified using integration by parts, and Formula 84
1
du 3D и — In(1 + en) + C
Formula 84
1 + eu
can be verified using substitution. Note that the integrals in Appendix B are classified
according to the form of the integrand. Several of the forms are listed below.
(a + bu)
Ja + bu
u"
(а + bu + cu?)
(a? + u?)
u?
Ju + a?
Ja?
Trigonometric functions
Inverse trigonometric functions
Exponential functions
Logarithmic functions
Transcribed Image Text: EXAMPLE 1
Integration by Tables
dx
Find
1
Solution Because the expression inside the radical is linear, you should consider
forms involving Ja + bu.
а
S:
a + bu
+ C
du
arctan
Formula 17 (a < 0)
a +
bu
- a
a
Let a = – 1, b
1, and u = x. Then du
dx, and you can write
dx
2 arctan /x – 1 + C.
1
EXAMPLE 2
Integration by Tables
•••D See LarsonCalculus.com for an interactive version of this type of example.
Find
9 dx.
Solution Because the radical has the form Ju? – a?, you should consider
Formula 26.
lu/u - a² – a Inlu + Vu – a²) + C
u?
a² du
Let u = x? and a = 3. Then du
2x dx, and you have
%3D
9 dx
V)2 – 32 (2x) dx
- 9 – 9 In? + JA* – 9) + C.
EXAMPLE 3
Integration by Tables
Evaluate
dx.
1 + e-x²
Solution Of the forms involving e", consider the formula
Si
du
%3D и — In(1 + е") + С.
Formula 84
1 + e"
Let u = -x². Then du =
- 2x dx, and you have
- 2x dx
dx
+
1 + e¬x?
- – In(1 + e¬*)] + c
x2
= [ + In(1 + e=*)) + c.
So, the value of the definite integral is
* + In(1 + e¬*³) | = 14 + In(1 + e-4) – In 2] = 1.66.
dx
1 + e
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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