**This question was asked at Monday’s tutorial (10/01/11) but the fire alarm went off mid-solution**

**Section 6.4, Q. 5**

*Evaluate the following integral:*

**Solution**

*(Remarks in italics are by me and would not be required in an exam situation)*

*Simplify the integrand to get it into a usable form:*

*Rule 1 (Section 6.4)*

*Given a rational function with , such that factors into non-repeated linear terms:*

*(non-repeated means that no linear term is equal to a constant multiple of another; e.g. for , )*

*Then*

*for some constants .*

Now

(*)

*At this point we have two options:*

*Multiply out and rearrange the right-hand side into a quadratic in . Compare and set equal the coefficients of , and the constant terms to generate three simultaneous equations for .**Choose three values of (say for ease of computation, see the note below) to generate**three simultaneous equations for . We’ll use this method here:*

,

,

,

Making the substitutions and (*it’s always a good idea to do this rather than court difficulty with terms of the form with *), and noting :

*Recalling , try to write this expression as for some functions of :*

*Now using , , and :*

** Note: **You may notice that in using technique 2. for finding that we use values of for which both sides of (*) are actually undefined. By means of the following general example let me show you why this is not a problem.

**Example**: *Let such that . Find the partial fraction decomposition of *

*Solution*: In essence this question asks us to find real numbers such that:

These expressions must agree at all points except at where is not defined. Suppose we multiply across by :

(**)

Now look at this equation in isolation, and forgetting about our original problem, we can see that we can find , such that this equation holds for all . It is an easy exercise to show that (**) is solved by: and . That is *for all *:

Now if we take this expression, and divide across by we will get a new expression,* which is true as long as we didn’t divide by zero, i.e. . *This statement is:

These functions agree on , as is our want.

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