Integration by partial fractions

A function of form P(x)/Q(x), where P(x) and Q(x) are polynomials, is known as rational function. Consider rational function

The 2 fractions on RHS are called as partial fractions. To integrate rational function on the LHS, it is enough to integrate the 2 fractions on the RHS, which are integrable easily. This is called as method of partial fractions. In case degree of P(x) (numerator) is not less than that of Q(x) (denominator), we carry out division of P(x) by Q(x) and reduce the degree of numerator.

To write P(x)/Q(x) in partial fractions, 1^st of all we write 

Q(x) = (x - a)^{k ...} (x² + ax + b)^r ... where the binomials are different, and then the set

here A₁, A₂, ..., A_k, M₁, M₂, ......, M_r, N₁,  N₂, ...... ,N_r are real  constants which are to be determined. These can be determined by reducing the both sides of the above identity to integral form and equating coefficients of equal powers of x, which gives the system of linear equations in the coefficient. (This method is called as method of comparison of coefficients). The constants can be obtained by substituting suitably chosen numerical values of x in both sides of identity.

Note:   Before proceeding to write a rational function as the sum of partial fractions, we should be taken to determine that it is either a proper rational fraction or can be rewritten as.

A rational function P(x)/Q(x) is proper if degree of polynomial Q(x) is greater than degree of the polynomial P(x). In case degree of P(x) greater than or equal to degree of Q(x), we 1^st write P(x)/Q(x) = h(x) + P(x)/Q(x), where  h(x) is a polynomial and p(x) is a polynomial  of degree less than degree of polynomial Q(x). 

Example:  Evaluate .

Solution:        Put sinx = t  => cosx dx  = dt

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