Calculus Of One Real Variable – By Pheng Kim Ving

10.1 
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1. Integrals And Integration 
The fundamental theorem of calculus, as presented in Section
9.4 Theorem 2.1, provides us a powerful method to
calculate definite integrals, by the formula:
where F
is any antiderivative of f.
This method places the burden of calculation squarely on finding an
antiderivative F of
f.
If an antiderivative of a function is known or found then definite
integrals of that function can readily be computed. So the
task is often simply to find antiderivatives, not definite integrals. An
antidifferentiable function f
has infinitely many
antiderivatives (any 2 of them differ from each other by a constant). Now the
general antiderivative of f
represents all the
antiderivatives of f.
Thus the task is actually simply to find general antiderivatives, when no
definite integral is needed at
the moment.
In Section
9.4 Definition 3.1 we saw that the general antiderivative is the indefinite
integral. Consequently in terms of
integrals the task is simply to find indefinite integrals, when no definite
integral is needed at the moment. If the indefinite
integral of a function is known or found, definite integrals of that function
can readily be calculated. Many known indefinite
integrals are recorded in integral tables for all to use.
The word “integral” refers to the definite integral or the
indefinite integral. Which one it refers to will be clear from the
context. Integration as defined in Section
9.4 Definition 4.1 is the process of finding a definite integral or an
indefinite
integral. Which process it is will be clear from the context.
Calculate:
Solution
EOS
Note that we don't need to insert a constant into the antiderivative (x^{3}/3) + e^{x}; see Section 9.4 Evaluation.
a. Find this derivative:
EOS
If the derivative of x^{2}e^{x} is
(2x + x^{2})e^{x}, then an
antiderivative of (2x
+ x^{2})e^{x} is x^{2}e^{x}, so the general
antiderivative or
indefinite integral of (2x
+ x^{2})e^{x} is x^{2}e^{x} + C, where C
is an arbitrary constant. Remark that it's not necessary to include
the constant C of the
indefinite integral in the computation of the definite integral; see Section
9.4 Evaluation.
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2. Integration By Inspection 
Evaluate:
We know this differentiation formula: (d/dx)
x^{2} = 2x, that is, the derivative of x^{2} is 2x. By inspection we see that 2x is
the derivative of x^{2}, so that an
antiderivative of 2x
is x^{2}. Thus the
general antiderivative or indefinite integral of 2x is x^{2} +
C, where C is an arbitrary constant. This is an example of
integration by inspection.
Find:
EOS
We use a trigonometric identity to render the integrand into
a form whose antiderivative is obvious. This is another
example of integration by inspection.
Integration by inspection refers to the situation
where we by inspecting the integrand see right away what its
antiderivative is, as in Example 2.1, or see that it can be rendered into a
form whose antiderivative is obvious, as in
Example 2.2. Integration by inspection clearly requires that we know
differentiation formulas and rules. For example we
know these differentiation formulas: (d/dx) x^{2} = 2x,
(d/dx) (1/2)x = 1/2, and (d/dx) sin x
= cos x
and the chain rule
(for (d/dx) sin 2x
= (cos 2x)(2)
= 2 cos 2x). Part 4 below gives a table of basic integrals corresponding
to some
differentiation formulas and rules.
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3. Integrals Requiring Integration Techniques 
Now consider this integral:
This is hard to be evaluated by inspection. There's a
technique to find it. There are integrals that require techniques. The
next several sections present various techniques to find integrals that are
hard to evaluate or can't be evaluated by
inspection. In this section we handle only integrals that can be evaluated by
inspection.
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The following integrals are some of the basic ones that derive from differentiation formulas and rules and that should be memorized. Each one can be verified by simply differentiating the righthand side to obtain the integrand on the lefthand side. Here are 2 examples:
For formula 1:
Some Basic Integrals
Evaluate:
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5. Integrals Of Some Operations On Functions 
Here are integrals of some operations on functions, namely of sum, difference, and constant multiple of functions:
where a and b are constants. 
These formulas can easily be proved by deriving them from differentiation formulas. For example for formula 1:
Calculate:
Solution
EOS
Problems & Solutions 
1. Calculate the following indefinite integrals.
Solution
2. Compute this indefinite integral:
Solution
3. Evaluate this indefinite integral:
Solution
4. Find:
Solution
5. Evaluate:
Solution
Note
We insert the constant C
only after the last integral has been evaluated. There's no point to insert it
before then, because
if we did, then we would have to add up those constants to get just a single
constant.
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