IIT JEE Maths Formula Sheet 2027

7.5 INTEGRATION BY PARTIAL FRACTIONS AND BY PARTS

A rational function is a quotient P(x)/Q(x) of two polynomials with Q(x) ≠ 0. When the degree of the numerator is smaller than that of the denominator the function is said to be proper; otherwise it is improper. An improper rational function can always be written, by ordinary long division, as a polynomial plus a proper rational function, so that the problem of integrating any rational function reduces to that of integrating a proper one.

A proper rational function whose denominator can be split into linear and quadratic factors may be expressed as a sum of simpler fractions, called partial fractions, each of which can be integrated directly. For example, a denominator of the form (x − a)(x − b) with a ≠ b leads to fractions A/(x − a) and B/(x − b), the constants A and B being found by comparing coefficients.

Another useful method applies when the integrand is a product of two functions. It follows from the product rule of differentiation and is stated as

The success of this method depends on a sensible choice of the first function; a common guide is the order inverse trigonometric, logarithmic, algebraic, trigonometric, exponential.