Trigonometric substitution: direct evaluation

4. Use the substitution \(x = 2 \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$

11. \begin{figure}[h] \end{figure}

1. By writing \(\sec \theta\) as \(\frac { 1 } { \cos \theta }\), show that when \(x = 3 \sec \theta\), $$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = 3 \sec \theta \tan \theta$$ Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { \sqrt { x ^ { 2 } - 9 } } { x } \quad x \geqslant 3$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 6\)
2. Use the substitution \(x = 3 \sec \theta\) to find the exact value of the area of \(R\). [Solutions based entirely on graphical or numerical methods are not acceptable.]

4. Use the substitution \(x = \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$

10

1. Use the substitution \(x = \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$
2. Find the exact value of $$\int _ { 1 } ^ { 3 } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x$$ 4

2 Use the substitution \(x = \tan \theta\) to find the exact value of $$\int _ { 1 } ^ { \sqrt { 3 } } \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } } \mathrm {~d} x$$

4 Use the substitution \(x = \frac { 1 } { 3 } \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 6 } } \frac { 1 } { \left( 1 - 9 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$

8

1. The curve \(y = \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }\) is shown in Fig. 8. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-08_495_1058_1105_315} \captionsetup{labelformat=empty} \caption{Fig. 8}
   \end{figure}
   1. Show that \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 20 x ^ { 2 } - 4 } { \left( 1 + x ^ { 2 } \right) ^ { 4 } }\).
   2. In this question you must show detailed reasoning. Find the set of values of \(x\) for which the curve is concave downwards.
2. Use the substitution \(x = \tan \theta\) to find the exact value of \(\int _ { - 1 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } d x\). Answer all the questions.
   Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

6

1. Given that \(x = \frac { 1 } { \sin \theta }\), use the quotient rule to show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta\).
   (3 marks)
2. Use the substitution \(x = \operatorname { cosec } \theta\) to find \(\int _ { \sqrt { 2 } } ^ { 2 } \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x\), giving your answer to three significant figures.
   (9 marks)

7

1. Given that \(y = \frac { \sin \theta } { \cos \theta }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec ^ { 2 } \theta\).
2. Given that \(x = \sin \theta\), show that \(\frac { x } { \sqrt { 1 - x ^ { 2 } } } = \tan \theta\).
3. Use the substitution \(x = \sin \theta\) to find \(\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\), giving your answer in terms of \(x\).

3. Use the substitution \(x = \sin \theta\) to show that, for \(| x | \leq 1\), $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { x } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } + c \text {, where } c \text { is an arbitrary constant. }$$

6. (a) Use the substitution \(x = 2 \sin u\) to evaluate $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$$ (b) Use integration by parts to evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x d x$$ 6. continued

Use the substitution \(2x = \tan \theta\) to find the exact value of $$\int_0^{\frac{1}{2}} \frac{12}{(1 + 4x^2)^2} \, dx .$$ Give your answer in the form \(a + b\pi\), where \(a\) and \(b\) are rational numbers. [9]

Use the substitution \(x = \sin \theta\) to find the exact value of $$\int_0^1 \frac{1}{(1-x^2)^{3/2}} dx.$$ [7]