Use trigonometric identity before integration

4

1. Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos 2 x \mathrm {~d} x = \frac { 1 } { 2 }\).
2. By using an appropriate trigonometrical identity, find the exact value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } 3 \tan ^ { 2 } x \mathrm {~d} x$$

6

1. Find \(\int 4 \mathrm { e } ^ { x } \left( 3 + \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
2. Show that \(\int _ { - \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 4 } \pi } \left( 3 + 2 \tan ^ { 2 } \theta \right) \mathrm { d } \theta = \frac { 1 } { 2 } ( 8 + \pi )\).

6
\includegraphics[max width=\textwidth, alt={}, center]{cc051d68-7e21-4dc1-b34d-6fb7f12a52fd-3_401_586_817_778} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.

1. Find the exact \(x\)-coordinate of \(P\).
2. Show that the equation of the curve can be written $$y = 5 + 8 \sin x - 2 \cos 2 x$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes.

6
\includegraphics[max width=\textwidth, alt={}, center]{3b217eb4-3bd3-4800-a913-749754bf109f-3_401_586_817_778} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.

1. Find the exact \(x\)-coordinate of \(P\).
2. Show that the equation of the curve can be written $$y = 5 + 8 \sin x - 2 \cos 2 x$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes.

7

1. Find \(\int \frac { 1 + \cos ^ { 4 } 2 x } { \cos ^ { 2 } 2 x } \mathrm {~d} x\).
2. Without using a calculator, find the exact value of \(\int _ { 4 } ^ { 14 } \left( 2 + \frac { 6 } { 3 x - 2 } \right) \mathrm { d } x\), giving your answer in the form \(\ln \left( a \mathrm { e } ^ { b } \right)\), where \(a\) and \(b\) are integers.

5
\includegraphics[max width=\textwidth, alt={}, center]{4c71f68a-efb9-4408-bf03-874e0d4426d5-2_458_807_1786_667} The diagram shows the curve $$y = 8 \sin \frac { 1 } { 2 } x - \tan \frac { 1 } { 2 } x$$ for \(0 \leqslant x < \pi\). The \(x\)-coordinate of the maximum point is \(\alpha\) and the shaded region is enclosed by the curve and the lines \(x = \alpha\) and \(y = 0\).

1. Show that \(\alpha = \frac { 2 } { 3 } \pi\).
2. Find the exact value of the area of the shaded region.

5

1. Find \(\int \left( 4 + \tan ^ { 2 } 2 x \right) \mathrm { d } x\).
2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \sin \left( x + \frac { 1 } { 6 } \pi \right) } { \sin x } \mathrm {~d} x\).

5

1. Prove the identity \(\cos 4 \theta - 4 \cos 2 \theta + 3 \equiv 8 \sin ^ { 4 } \theta\).
2. Using this result find, in simplified form, the exact value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 4 } \theta \mathrm {~d} \theta$$

4

1. Find \(\int \tan ^ { 2 } 3 x \mathrm {~d} x\).
2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 3 x } + 4 } { \mathrm { e } ^ { x } } \mathrm {~d} x\). Show all necessary working.

7

1. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \operatorname { cosec } ^ { 2 } x \mathrm {~d} x = \sqrt { } 3\). By using appropriate trigonometrical identities, find the exact value of
3. \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { 2 } x \mathrm {~d} x\),
4. \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 1 - \cos 2 x } \mathrm {~d} x\).

4

1. Find \(\int \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\).
2. Express \(\sin ^ { 2 } 3 x\) in terms of \(\cos 6 x\) and hence find \(\int \sin ^ { 2 } 3 x \mathrm {~d} x\).

3

1. Find \(\int 4 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta\).
2. Find the exact value of \(\int _ { - 1 } ^ { 6 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x\).

5

1. Find \(\int \left( \tan ^ { 2 } x + \sin 2 x \right) \mathrm { d } x\).
2. Find the exact value of \(\int _ { 0 } ^ { 1 } 3 \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\).

2
\includegraphics[max width=\textwidth, alt={}, center]{8b051aee-4920-42a0-8b74-cbfa9f3c1ab1-2_429_821_826_662} The variables \(x\) and \(y\) satisfy the equation \(y = K x ^ { p }\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( 1.28,3.69 )\) and \(( 2.11,4.81 )\), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places.

1. Find \(\int \tan ^ { 2 } 4 x \mathrm {~d} x\).
2. Without using a calculator, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } ( 4 \cos 2 x + 6 \sin 3 x ) \mathrm { d } x\).

4

1. Find \(\int \frac { 4 + \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } \mathrm {~d} \theta\).
2. Given that \(\int _ { 0 } ^ { a } \frac { 2 } { 3 x + 1 } \mathrm {~d} x = \ln 16\), find the value of the positive constant \(a\).

6

1. Show that \(\int _ { 1 } ^ { 6 } \frac { 12 } { 3 x + 2 } \mathrm {~d} x = \ln 256\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( 8 \sin ^ { 2 } x + \tan ^ { 2 } 2 x \right) \mathrm { d } x\), showing all necessary working.

6

1. Show that \(\int _ { 2 } ^ { 18 } \frac { 3 } { 2 x } \mathrm {~d} x = \ln 27\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 4 \sin ^ { 2 } \left( \frac { 3 } { 2 } x \right) \mathrm { d } x\). Show all necessary working.

5

1. Find \(\int \left( \tan ^ { 2 } x + \sin 2 x \right) \mathrm { d } x\).
2. Find the exact value of \(\int _ { 0 } ^ { 1 } 3 \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\).

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = ( 1 + 2 \cos 2 x ) ^ { 2 }$$

1. Show that $$( 1 + 2 \cos 2 x ) ^ { 2 } \equiv p + q \cos 2 x + r \cos 4 x$$ where \(p , q\) and \(r\) are constants to be found. The curve touches the positive \(x\)-axis for the second time when \(x = a\), as shown in Figure 4. The regions bounded by the curve, the \(y\)-axis and the \(x\)-axis up to \(x = a\) are shown shaded in Figure 4.
2. Find, using algebraic integration and making your method clear, the exact total area of the shaded regions. Write your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-32_2255_51_313_1980}

1. (a) Using the identity for \(\cos ( A + B )\), prove that

$$\cos 2 A \equiv 2 \cos ^ { 2 } A - 1$$ (b) Hence, using algebraic integration, find the exact value of $$\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 8 } } \left( 5 - 4 \cos ^ { 2 } 3 x \right) d x$$

8. (i) Find \(\int \tan ^ { 2 } x \mathrm {~d} x\).
(ii) Show that $$\int \tan x \mathrm {~d} x = \ln | \sec x | + c$$ where \(c\) is an arbitrary constant.
\includegraphics[max width=\textwidth, alt={}, center]{1e93a786-6105-4c69-a79a-a5f6e6c4aa0a-2_554_784_1484_507} The diagram shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \tan x\).
The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 3 }\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
(iii) Show that the volume of the solid formed is \(\frac { 1 } { 18 } \pi ^ { 2 } ( 6 \sqrt { 3 } - \pi ) - \pi \ln 2\).

2．Given that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( 1 + \tan \left[ \frac { 1 } { 2 } x \right] \right) ^ { 2 } \mathrm {~d} x = a + \ln b$$ find the value of \(a\) and the value of \(b\) ．

5. \begin{figure}[h] \end{figure} Show that the area of the finite region between the curves \(y = \tan ^ { 2 } x\) and \(y = 4 \cos 2 x - 1\) in the interval \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), shown shaded in Figure 3, is given by $$2 \sqrt { 2 \sqrt { 3 } } - 2 \sqrt { 2 \sqrt { 3 } - 3 }$$

3 By expressing \(\cos 2 x\) in terms of \(\cos x\), find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \cos 2 x } { \cos ^ { 2 } x } \mathrm {~d} x\).

5

1. Show that \(\frac { 1 } { 1 - \tan x } - \frac { 1 } { 1 + \tan x } \equiv \tan 2 x\).
2. Hence evaluate \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 6 } \pi } \left( \frac { 1 } { 1 - \tan x } - \frac { 1 } { 1 + \tan x } \right) \mathrm { d } x\), giving your answer in the form \(a \ln b\).