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

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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:

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:

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:

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:

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:

Finding the antiderivative of f(x) will produce:

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

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):

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

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 2: Exponents

Property 3: Logarithms

Property 4: Exponential Functions

Property 5: Arctan Function

Property 6: Arcsin Function

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.

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

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

We note the function and it's antiderivative below:

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

We note the function and it's antiderivative below:

Therefore, we get the following result:

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

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

Recall that the derivative of e^x is itself so the antiderivative produces the same result.

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

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:

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:

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