partial fraction expansions

Any proper rational function can be expressed as sum of rational fractions, each having a factor of Q(x). Each such fraction is known as Partial fraction..

Before using this technique of integration, let us recall what we have learnt about partial fraction..

Rational function: If P(x) and Q(x) are two polynomials in x, then the ratio of two polynomials, P(x) / Q(x) is called a rational function, where Q(x) is not equal to zero. Proper rational function: If the degree of the numerator of the rational function is less than that of the den..

For n Q, we have: ..

Which of the following partial fractions represents the given rational function 6 x ² - 9 ? => 1 x + 3 - 1 x + 3 or - 1 x + 3 + 1 x - 3 or 1 x + 3 or 1 x + 3 + 1 x - 3..

For any rational number n, We accept this expansion without proo..

Particular Terms for Fractional Index - Sometimes, a particular term satisfying certain conditions is required in the binomial expansion of the type (1+x) n . This can be done by expanding (1+x) n to certain terms and then locating the required term. Generally this becomes a ted..

Binomial Theorem for Fractional Index - For any rational number n, We accept this expansion without proof. The restriction on x is not required when n is a natural number. Now, we shall see that when n is a natural number, then the above expansion coincides with that a..

In the partial separation of heavy water from ordinary water, advantage is taken of the small difference in the boiling points of protium oxide (373.2 K) and deuterium oxide (374.3 K). Since the difference in boiling points is very small, a long fractionating column (about 13m)..

Which of the following partial fractions represents the given rational function x - 9 x ( x + 9 ) ? => 1 x + 2 x + 9 or -1 x + 2 x - 9 or 2 x + - 1 x + 9 or -1 x + 2 x + 9..