Mathematics

Problem

Define

K(a)=\int x\sqrt{x^{2}+a^{2}}\,dx,\quad a>0.

Answer the following questions. a. Part 1: (i) Find a simplified closed-form for

K(a)

, writing the result as a function of

x

a

and a constant of integration

C

. b. Part 2: (ii) Keeping

x

fixed, differentiate the expression obtained in part (i) partially with respect to

a

and simplify the derivative. c. Part 3: (iii) Hence or otherwise, obtain an antiderivative of

\dfrac{a}{\sqrt{x^{2}+a^{2}}}

with respect to

x

. Compare your answer with the standard result.