For a positive constant

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

a

, consider the integral

I(x,a)=\displaystyle\int \frac{dx}{\left(x^{2}+a^{2}\right)^{2}}.

Answer the following. (i) Evaluate

I(x,a)

and simplify your result as a single expression involving

x

a

and an arbitrary constant of integration

C

. (ii) Treating

a

as a variable, differentiate the antiderivative obtained in part (i) partially with respect to

a

. Show that the result equals

\frac{\partial}{\partial a}\Bigl[\,I(x,a)\Bigr]=\int \frac{\partial}{\partial a}\Bigl[\,(x^{2}+a^{2})^{-2}\Bigr]dx.