Using Partial fractions

This chapter explores partial fractions. The objective for this chapter is to integrate a fraction using partial fractions. Before attempting the chapter you must have prior knowledge of partial fractions, and integrating simple fractions using natural logs.

Using partial fractions – simple linear

We can use the knowledge that we have about partial fractions to integrate fractions for example finding the following integral
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First we shall rewrite the fraction as partial fractions as shown below;
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…substitute in the know values…
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Now we can equate the top lines;
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Let x=2, so we get;
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Next we simplify;
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Now we know that;
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Next let x=-1
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So the integral becomes;
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Now we can integrate each fraction using natural logs (ln) we get;
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Next we simplify to get;
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Another example

Here is another example, find the following integral
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To simplify we first factorise the bottom to get;
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Now we let;
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Multiply through by x(2x-1)
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…and we get;
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Now we let x=1/2;
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…we get;
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Now let x=0;
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…we get;
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So now the integral becomes;
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Partial Fractions – repeated denominator

We can also use partial fractions for repeated denominators as well, for example integrate the following;
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This time we stop the partial fractions differently, let;
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Next we write over the common denominator…
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Now we can equate the top lines;
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Let x=-1/2 and substitute in;
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Now we know;
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Next we let x=-2;
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…we know that;
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And now we can compare the coefficients of x².
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…we now know that;
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So the integral becomes;
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We can use logs for the first and last terms. We need to solve for the second term;
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…we can rewrite it as;
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…we can now guess the result as being;
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…we check whether it is correct by differentiating…
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The differential is almost right we have to adjust the factor of 2 as shown below…
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So the final integral becomes;
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…we simplify…
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Improper fractions

You may have to work with fractions that are improper fractions, for example integrate the following;
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…we can rewrite the fraction in the form;
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Let;
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…multiply through to get;
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Let x = -1
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…we know that;
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Let use let x=2;
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…we now know that;
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…when we compare the x² terms we get;
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So the integral becomes;
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