For the rational function R(x)= 5x-2 (x-1)(2x+3)

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

For the rational function

R(x)=\frac{5x-2}{(x-1)(2x+3)},

consider the indefinite integral

\displaystyle\int R(x)\,dx

. (a) Show algebraically that the denominator of

R(x)

factors into two distinct linear terms and explain why this guarantees that

\int R(x)\,dx

can be rewritten as a combination of logarithmic functions plus a constant. (b) Without determining any numerical values, express

R(x)

in the form

\frac{A}{x-1}+\frac{B}{2x+3},

where

A

and

B

are real constants to be found later. (c) Describe in writing how the values of

A

and

B

influence the coefficients appearing in the eventual logarithmic antiderivative.