Rules of Integration for Indefinite
Integral
In order to simplify the task of finding
integrals, some integration techniques are
used to help finding integrals by converting the integrand into some simple
integrands.
Method of Partial Fractions for
Indefinite Integral
The technique of
Integration by Partial Fractions is method of decomposing the original
integrand of the rational function format into the sum of some simple algebraic
fractions with its denominator becomes simple polynomial function. The strategy is to
break down the calculation work from determine the integral of a rational
function with complex format into the sum of a finite number of integrals of
simple algebraic fractions. Therefore the integrand of the integral is expressed
in partial fractions before proceeding to determine the value of the integral of
the orginal integrand. Imply
where Φ(x) and φ(x) are rational, integral, algebraical functions of x
Partial Fraction Decomposition:
The method of partial fraction decomposition can be applied to all rational
functions with the degree of its numerator is less than the degree of its denominator.
Therefore polynomial long division may be used to resolve the problem of
improper fractions. imply
where Q(x) is quotient polynomial, R(x) is remainder polynomial with degree less
than the degree of the denominator φ(x). And with the nominator Φ(x) is the dividend and the
denominator φ(x) is the divisor, i.e dividend=divisor x quotient +
remainder.
Denominator Factorization
After converting the rational function of the integrand to a proper fraction for
partial fractions, the polynomial of the denominator can then be factored
completely, that is no factoring of the polynomial of the denominator can be
carried further. The factorized denominator is a product of either be monomial or
binomial of the linear
form ax+b or irreducible trinomial of the quadratic form ax2+bx+c.
An irreducible trinomial is a prime if and only if the quadratic has no real
roots. When the quadratic has a root x1, then xx1 must be the factor of the quadratic.
An irreducible trinomial can be checked using the quadratic formula:
Therefore when b24ac>=0, the roots are real numbers
and the trinomial can be factored. And thus the trinomial is irreducible if and
only if the discriminant b24ac<0.
The integrand after denominator factorization is
Method of Factorization
Methods to examine and to determine the factors of a polyonomial are:

Factor out the greatest common factor of the polyonomial by determining the
largest monomial that can divide each term of the polynomial and removing out
the common factor from the original polyonomial.

Identify common factors of the remaining polynomial by grouping terms together
in a regular pattern.

scanning for common binomial if the number of terms in the polynomial is even.

Group terms in the polynomial into pairs with a common factor that is to group
two terms in the polynomial with a regular pattern so that the
degree difference of terms in a group is the same for all groups and the
difference between paried terms is the monomial factors only.
Therefore the polynomial can be expressed as

Factor out the greatest common factor of each individual grouped binomial by
determining the largest monomial that can divide each term of the individual
grouped binomial seperately.

Factor out the common binomial

Repeat factoring
out the common binomial from the remaining polynomial whenever possible.

scanning for common
trinomial if the number of terms in the polynomial is odd and greater or equal
to three.

If the number of terms in the polynomial is equal to three, try the standard factor trinomial factor into factors
first so as to factor as a product of two binomial.

Applying the technique of reverse FOIL method, e.g. LastInnerOuterFirst
product, to determine the common trinomial factors, i.e. prime or irreducible quadratic polynomials.

Factor standard binomial factor into factors

form a2b2
Therefore the degree of binomial is multiple of two with coefficients of
opposite sign can be applied. Repeat factoring whenever possible.

form a3b3
Therefore the degree of binomial is multiple of three with coefficients of
opposite sign can be applied. Repeat factoring whenever possible.

form a3+b3
Therefore the degree of binomial is multiple of three with coefficients of
equal sign can be applied. Repeat factoring whenever possible.

form a4b4
This is the special case of type a. Repeat factoring whenever possible.

Factor
standard polynomial factor into standard binomial factors.
Sometimes the polynomial can be factorized into product of
binomial factors.
Following are some standard polynomial pa4+qa3+ra2+sa+t for reference.

Factor trinomial factor into
product of binomial factors

standard
form a22ab+b2 or standard
format ra2+sa+t
Therefore
test for the squares and product of coefficients, the
standrad trinomial, a perfect square trinomial, can be factored to the square of
a binomial. Repeat factoring whenever possible.

standard
form a2+ab+b2 or standard
format ra2+sa+t
Therefore test for the squares and product of coefficients, the
standrad trinomial, a perfect square trinomial, can be factored to the square of
a binomial. Repeat factoring whenever possible.

form a2+ca+d or standard format ra2+sa+t
Therefore
the coefficient of the highest degree of the tribomial is eqaul to 1. The trinomial
can be transformed to a four terms polynomial for grouping factorization by
spliting the middle variable c into two terms. Or the trinomial can be factored
by trial and error by factoring the constant d to all possible product pair set and testing
the sum of each set with the coefficient c. Repeat factoring whenever possible.

form ba2+ca+d or standard format ra2+sa+t
Therefore
the coefficient of the highest degree of the tribomial is greater than 1. The trinomial
can be transformed to a four terms polynomial for grouping factorization by
spliting the middle variable c into two terms. Or the trinomial can be factored
by trial and error by factoring the constant d to all possible product pair set and testing
the sum of each set with the coefficient c. Repeat factoring whenever possible.

Factor standard quadrinomial factor into
product of binomial factors.

form a33a2b+3ab2b3
or standard polynomial qa3+ra2+sa+t

form a3+3a2b+3ab2+b3
or standard polynomial qa3+ra2+sa+t
Therefore the coefficient of the highest degree of a standard quadrinomial is 3
with same coefficient format can be applied. Repeat factoring whenever possible.enever possible.

Factor standard quintinomial factor into
product of binomial factors.

form a43a2b+3ab2+3ab2b3
or standard polynomial pa4+qa3+ra2+sa+t

form a3+3a2b+3ab2+b3
or standard polynomial pa4+qa3+ra2+sa+t
Therefore the coefficient of the highest degree of a standard quadrinomial is 3
with same coefficient format can be applied. Repeat factoring whenever possible.enever possible.

Factor
standard polynomial factor into standard trinomial factors.
Sometimes the polynomial can be factorized into product of trinomial factors.
Following are some standard polynomial pa4+qa3+ra2+sa+t for reference.

Factor trinomial factor into standard trinomial factors.
form pa4ra2+t ; b2>2c

Factor quadrinomial factor into standard trinomial factors.

form pa4+ra2+sa+t

form pa4+qa3+ra2t

Factor quintinomial factor into standard trinomial factors.

form pa4+qa3+ra2+sa+t

form pa4qa3+ra2+sa+t ; b2>2c

form pa4qa3+ra2sa+t

form pa4+qa3+ra2sa+t ; b2>2c