Mathematics

Problem

Let

k

be a non-zero real parameter and consider the integral

I(k)=\int \frac{k x+1}{x^{2}+2k x+k^{2}+1}\,dx.

(a) Show that the denominator can be expressed in the form

(x+k)^{2}+1

. (b) Hence obtain a simplified exact expression for

I(k)

in terms of elementary functions of

x

and the parameter

k