Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 10: Techniques Of Integration – Section 10.1: Integration By Inspection

 

10.1
Integration By Inspection

 

 

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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.

 

Example 1.1

 

Calculate:

 

 

Solution

EOS

 

Note that we don't need to insert a constant into the antiderivative (x3/3) + ex; see Section 9.4 Evaluation.

 

Example 1.2

 

a. Find this derivative:

 

 

Solution

EOS

 

If the derivative of x2ex is (2x + x2)ex, then an antiderivative of (2x + x2)ex is x2ex, so the general antiderivative or
indefinite integral of (2x + x2)ex is x2ex + 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

 

Example 2.1

 

Evaluate:

 

 

Solution

EOS

 

We know this differentiation formula: (d/dx) x2 = 2x, that is, the derivative of x2 is 2x. By inspection we see that 2x is
the derivative of x2, so that an antiderivative of 2x is x2. Thus the general antiderivative or indefinite integral of 2x is x2 +
C, where C is an arbitrary constant. This is an example of integration by inspection.

 

Example 2.2

 

Find:

 

 

Solution

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) x2 = 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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4. Some Basic Integrals

 

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 right-hand side to obtain the integrand on the left-hand side. Here are 2 examples:

 

For formula 1:

 

 

Some Basic Integrals

 

 

 

 

Example 4.1

 

Evaluate:

 

 

Solution

EOS

 

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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:

 

 

Example 5.1

 

Calculate:

 

 

Solution

EOS

 

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Problems & Solutions

 

1. Calculate the following indefinite integrals.

 

  

 

Solution

 

 

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2. Compute this indefinite integral:

 

   

 

Solution

 

 

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3. Evaluate this indefinite integral:

 

  

 

Solution

 

 

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4. Find:

 

  

 

Solution

 

 

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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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