Derivative then integrate by parts

8 \includegraphics[max width=\textwidth, alt={}, center]{326d0ea0-8060-4439-8043-3301b281a30f-3_391_826_1946_657} The diagram shows the curve \(y = x \cos \frac { 1 } { 2 } x\) for \(0 \leqslant x \leqslant \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y + 4 \sin \frac { 1 } { 2 } x = 0\).
2. Find the exact value of the area of the region enclosed by this part of the curve and the \(x\)-axis.

6．Figure 1 shows a sketch of part of the curve with equation \(y = x \sin ( \ln x ) , x \geqslant 1\) \begin{figure}[h] \end{figure} For \(x > 1\) ，the curve first crosses the \(x\)－axis at the point \(A\) ．
（a）Find the \(x\) coordinate of \(A\) ．
（b）Differentiate \(x \sin ( \ln x )\) and \(x \cos ( \ln x )\) with respect to \(x\) and hence find $$\int \sin ( \ln x ) \mathrm { d } x \text { and } \int \cos ( \ln x ) \mathrm { d } x$$ （c）（i）Find \(\int x \sin ( \ln x ) \mathrm { d } x\) ．
（ii）Hence show that the area of the shaded region \(\boldsymbol { R }\) ，bounded by the curve and the \(x\)－axis between the points \(( 1,0 )\) and \(A\) ，is $$\frac { 1 } { 5 } \left( \mathrm { e } ^ { 2 \pi } + 1 \right)$$

8

1. State the derivative of \(\mathrm { e } ^ { \cos x }\).
2. Hence use integration by parts to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos x \sin x \mathrm { e } ^ { \cos x } \mathrm {~d} x$$

4

1. Differentiate \(\mathrm { e } ^ { x } ( \sin 2 x - 2 \cos 2 x )\), simplifying your answer.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \mathrm { e } ^ { x } \sin 2 x \mathrm {~d} x\).

The curve \(C\) has equation \(y = 2 \sec x\), for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). Show that the arc length \(s\) of \(C\) is given by $$S = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 2 } x - 1 \right) d x$$ Find the exact value of \(s\). The surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\). Show that

1. \(S = 4 \pi \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 3 } x - \sec x \right) \mathrm { d } x\),
2. \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x \tan x ) = 2 \sec ^ { 3 } x - \sec x\). Hence find the exact value of \(S\).

6

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 1 } ^ { 5 } \ln x \mathrm {~d} x\), giving your answer to three significant figures.
2. 1. Given that \(y = x \ln x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence, or otherwise, find \(\int \ln x \mathrm {~d} x\).
   3. Find the exact value of \(\int _ { 1 } ^ { 5 } \ln x \mathrm {~d} x\).