Chapter 24 - Techniques of Integration

We have studied a great variety of functions and their derivatives. Until now the process of differentiation has been straight-forward and mechanical. For example

If in any of the above functions, x was replaced by u, where u is another function of x, then by the Chain rule:

From the product rule, the derivative of any product uv, the derivative was:

The rules for Integration, as we already know are unfortunately not so straightforward as in differentiation.  The reason for this is because there are numerous ways to write a function, yet only one of these ways will resemble a differentiable form. Therefore the key to integration lies in finding that integratable form and as we shall soon see, that form is not necessarily the simplest form.

Let us take a look at two simple functions and show the variety of ways it can be simplified, expanded and re-written.

or another example:

A reader pointed out an error in this last example. Two of the functions on the right are not equal to the left one ( fourth and sixth one from the top ).

Even though all these functions are equal to the original function, they can not be integrated directly as they do not appear in a differentiable form. The only two functions that can be integrated directly are the original functions , whose integrals are and respectively. If you closely examine the physically different, yet mathematically equal functions, you can notice that in most cases the integrable form is quite evident  and can easily be found with a few quick simplifications such as:

In other cases the distinction is not so evident.

This final and important chapter is devoted to the study of functions that at first glance look impossible to integrate; however, after the application of certain simplifications and re-writing techniques, the function can be integrated.

The first of these techniques is called integration by parts. It allows us to evaluate the integral of a product, in terms of the integral of one factor multiplied by the derivative of the other. From the product rule we have.

or just:

Integrating both sides gives us:

Solving for gives us:

For example to integrate the product of

Now that we have expressed the derivative as a sum of , we can integrate the derivatives to get:

The reason why this integration technique works is based on the product rule which allows us to differentiate a product as a sum of two functions, . Even when not integrating a product, one can always let such that v=x.

The next technique that we shall study is based on the derivatives for the Sine and inverse Sine function. From the previous chapter we saw how was defined as:

or by the definition of the integral the derivative of is . Based on this the derivative of Sin(x) was found to be Cos(x).

Furthermore we showed how all other trigonometric function such as Tan(x) and Sec(x) could be expressed in terms of Sin(x) and Cos(x), or just Sin(x) since Cos(x) is also dependent on Sin(x). With this in mind let us see how we can express the inverse tangent ans secant functions in terms of the inverse sine function.

The function is defined as the arclength from 0 to x on a circle or:

If we define the inverse tan function to give us the ratio of the x value to the y value for a particular arclength then:

Where gives us an arclength for that particular value of x, and the tangent of that arclength gives us . If we let and then differentiate both sides we get:

We have found the derivative of but we need to find the derivative of not . To do so we first need to define in terms of . The function tell us the x value for a given arc length. Now if for the same arclength , the Tangent were x, then what would be the Sine of the Arclength. To answer this, let us consider the following graph again: Error: The equations for square root of 1 - x^2 should all be 1 + x^2 in the next few lines.

In this circle, the tangent of the angle is x, and the Sine of the angle is . Hence:

For any angle whose Tangent is x, its sine will be . Differentiating both sides yields:

Further simplification yields:

The derivative of the inverse Tangent function is just or more importantly.

By similar means we can find the derivative of other inverse trigonometric functions, by first defining it in terms of the Sine of the angle and then differentiating it.

The underlying concept in these definitions is the fact that the inverse tangent, secant and any other trigonemetric functions can be expressed in terms of such that integrating them is much easier as we already have them defined.

As we mentioned before, the difficulty of integration lies in the fact that a function could be written in so many different ways.