5.In this question u and v are functions of $x$ .Given that $\int \mathrm { u } \mathrm { d } x , \int \mathrm { v } \mathrm { d } x$ and $\int \mathrm { uv } \mathrm { d } x$ satisfy

$$\int \text { uv } \mathrm { d } x = \left( \int \mathrm { u } \mathrm {~d} x \right) \times \left( \int \mathrm { v } \mathrm {~d} x \right) \quad \text { uv } \neq 0$$
\begin{enumerate}[label=(\alph*)]
\item show that $1 = \frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } + \frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }$

Given also that $\frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } = \mathrm { sin } ^ { 2 } x$,
\item use part(a)to write down an expression,in terms of $x$ ,for $\frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }$ ,
\item show that

$$\frac { 1 } { \mathrm { u } } \frac { \mathrm { du } } { \mathrm {~d} x } = \frac { 1 - 2 \sin x \cos x } { \sin ^ { 2 } x }$$
\item hence use integration to show that $\mathrm { u } = A \mathrm { e } ^ { - \cot x } \operatorname { cosec } ^ { 2 } x$ ,where $A$ is an arbitrary constant.
\item By differentiating $\mathrm { e } ^ { \tan x }$ find a similar expression for v .
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2013 Q5 [15]}}