8
\includegraphics[max width=\textwidth, alt={}, center]{88f67166-7b44-4b04-b323-f43827531495-3_558_1047_950_550}
The diagram shows a sketch of the curve \(y = x ^ { \frac { 1 } { 2 } } \ln x\) and its minimum point \(M\). The curve cuts the \(x\)-axis at the point \(( 1,0 )\).
- Find the exact value of the \(x\)-coordinate of \(M\).
- Use integration by parts to find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 4\). Give your answer correct to 2 decimal places.
- Express \(\frac { 10 } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\) in partial fractions.
- Hence, given that \(| x | < 1\), obtain the expansion of \(\frac { 10 } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.