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Basic Integration

Tutorial 1


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Topics
What is integration?
Basic Integration
Solving Indefinite Integration
Properties of Indefinite Integration
Examples of Indefinite Integration
Examples
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
Comments
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What is Integration?

In calculus, integration is used for finding the areas under a curve on a given graph. There is a lot more to it than that as you will see as these tutorials move forward.

One rule of thumb that you need is to know how to use anti-derivatives. This was reviewed in the previous section of this manual. Here is a list of very common derivatives of various functions used in calculus:

part 1

It is interesting to note that all trig functions that begin with a "c" above, have a negative derivative. This trick may help you out quite a bit when trying to determine the derivative in an integral.

Basic Integration

The symbol for integration is a large stretched out S. It may also contain a lower or upper bound which signify the area you want to search for. Here is a general formula for integration:

part 2

which means on the interval [a, b], integrate the function f(x). The dx is merely a notation. It stands for differentaion in practical terms.

So, what exactly is integration doing? Its purpose is to find the area between points a and b of a curve on a graph. This (the integration) will be an approximation of the area, not the exact area.

There might also be something called an indefinite integral as denoted here:

part 3

This means a general integral and is not looking for a particular area under a curve. You will see examples of each of these as we proceed.

Solving Indefinite Integration

An indefinite integral is one where you are not looking for a particular area under a curve but an antiderivative of a given function. Here is the form for an indefinite integral:

part 4

The above F(x) means the antiderivative of the function f(x). The C stands for any constant because the derivative of F(x) is the original function so the constant term, whatever it is, will disappear. Here is an example:

part 5

Finding the antiderivative of f(x) will produce:

part 6

And now to reverse it, the derivative of F(x) will be:

part 7

And no matter what constant we add to the antiderivative, it will go away when we take the derivative of the antiderivative. All of the below mean the exact same thing for the given function f(x):

part 8

When we take the derivative of the F(x) according to each of the corresponding letters above, we get:

part 9

So to summarize, all an indefinite integral is asking for is to find an antiderivative to a given function.

Properties of Indefinite Integration

There are some useful properties of indefinite integrals:

Property 1: Constants

property 1

Property 2: Exponents

property 2

Property 3: Logarithms

property 3

Property 4: Exponential Functions

property 4

Property 5: Arctan Function

property 5

Property 6: Arcsin Function

property 6

Properties 3, 5 and 6 will be shown in greater detail when we get to the techniques of integration. For now, we will see some general examples of indefinite integration.

Examples of Indefinite Integration

Here are a few examples of indefinite integration as some relate to the above properties.

Example 1

example 1a

This clearly shows property 1 above so the final answer is:

example 1b

Example 2

example 2a

This clearly shows property 2 above so the final answer is:

example 2b

Example 3

example 3a

We note the function and it's antiderivative below:

example 3b

We can do some cleaning up in the above antiderivative and get an answer of:

example 3c

Example 4

example 4a

We note the function and it's antiderivative below:

example 4b

Therefore, we get the following result:

example 4c

The above example now leads to some integrals that can be found easily just by using basic properties of antiderivatives. These are listed below:

part 10

Example 5

example 5a

We note the above property 4 and get a final result of:

example 5b

Recall that the derivative of ex is itself so the antiderivative produces the same result.

Example 6

example 6a

Making use of another property above, we get a result of:

example 6b

Example 7

example 7a

This example may seem tricky at first but recall that there is still a constant term in the exponent; here it is a -1. So therefore, use the same property as the above and get a result of:

example 7b

Example 8

example 8a

I can probably bet you said "how on earth do I solve this one?" Well, just look at one of the above properties for the arctan function. This is actually in the form of an arctan with the value of a being 3 (3 squared produces 9). You get answer of:

example 8b


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