Now to calculate ʃ dx where P(x)/Q(x) is a proper rational function, we first convert it into partial faction suing the method as shown in the given figure and then integrate using standard formula. So, if P(x)/Q(x) is improper, then P(x)/Q(x) = T(x) + P 1(x)/Q(x) Integrate both sides and rearrange, to get the integration by parts formula. f(x)/g(x) f(x)(g(x))(-1) or in other words f or x divided by g of x equals f or x times g or x to the negative one power. When we use this formula, we 'divide the integral in parts'. Many people use the letters u and v instead of f and g. This formula is called the integration by parts formula. We can convert an improper rational function into proper rational function by division method. Falling back on partial fractions (not as an integration technique) and using some reduction formula at some point seems inescapable. Using just the product rule we obtained an interesting formula for integration. If degree of P(x) is less than degree of Q(x), then it is called proper rational function otherwise it is improper. We know that a rational function is the ratio of two polynomials in the form P(x)/Q(x), where P ( x) and Q( x) are polynomials in x and Q( x) ≠ 0.