L&T, 15.31-43
We now study the integral , i.e., linear over quadratic, where the quadratic does not factorize.
Use this to rearrange the numerator into form
i.e., as a constant times the derivative of the denominator plus another constant. We can now split the integral,
The first integral on the r.h.s. can be done using the substitution ,
separately. The technique used is based on completing the square, , which after the substitution leads to a standard integral
Depending on the sign we get either an inverse tangent or a ratio of logarithms,
Example 5.18:
Evaluate .
Solution:
Now complete the square for the denominator, and find that
Substitute , ,
Thus we find
L&T, 1.3.3.5
Completing the square is a simple idea that is surprisingly useful. First a definition:
Let us look at a few examples:
polynomial | degree | |
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(a) | 1 | Also called linear, since if we plot |
(b) | 1 | the functions , , etc. |
(c) | 1 | we get a straight line |
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(d) | 2 | |
(e) | 2 | (also known as quadratic) |
(f) | 2 | |
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(g) | 3 | cubic |
(h) | 6 | |
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A polynomial of infinite degree is usually called an infinite power series.
Any polynomial of degree 2, i.e., a quadratic, can always be rearranged to have the form , as the square of a linear term plus a constant. Bringing a quadratic polynomial to this form is called completing the square.
“Completing the square” is bringing a quadratic to the form .
In general, if two polynomials are equal, it means that the coefficient of each power of the variable are equal, since each power varies at a different rate with the variable. So in order to complete the square, we must equate coefficients of powers of on both sides. We shall do this by example.
Now equate coefficients of
on both sides. We find
, or
. Then compare the coefficients of
. We conclude
. Using
we find
. Now equate the constant term,
. We conclude that
.
Collecting all the results we
find
: | , | ||
: | , | therefore | , |
const: | , | therefore | . |
It is often useful to write the constant as