Derivative then integrate by parts

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 \(A\) .

1. Find the \(x\) coordinate of \(A\) .
2. Differentiate \(x \sin ( \ln x )\) and \(x \cos ( \ln x )\) with respect to \(x\) and hence find $$\int \sin ( \ln x ) \mathrm { d } x \text { and } \int \cos ( \ln x ) \mathrm { d } x$$
3. 1. Find \(\int x \sin ( \ln x ) \mathrm { d } x\) .
   2. 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 \(\int _ { 1 } ^ { 5 } \ln x \mathrm {~d} x\).

\includegraphics{figure_8} The diagram shows the curve \(y = x\cos\frac{1}{2}x\) for \(0 \leqslant x \leqslant \pi\).

1. Find \(\frac{dy}{dx}\) and show that \(4\frac{d^2y}{dx^2} + y + 4\sin\frac{1}{2}x = 0\). [5]
2. Find the exact value of the area of the region enclosed by this part of the curve and the \(x\)-axis. [5]