For a fixed real parameter

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

Problem

k

, define the rational function

R_k(x)=\frac{k x+1}{x^{2}+5x+6}.

Throughout this question, assume

x\neq -2

and

x\neq -3

. (a) Write

R_k(x)

in the form

\displaystyle \frac{A(k)}{x+2}+\frac{B(k)}{x+3}

and state the resulting identity after clearing denominators. (b) Derive explicit formulas for

A(k)

and

B(k)

in terms of

k

. (c) Using your formulas, compute

A

and

B

when

k=4

. (d) Evaluate

\displaystyle \int R_k(x)\,dx

in terms of

k

, including the constant of integration.