Method of differences

8. $$I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0$$

1. Show that, for \(n \geqslant 1\)
2. Hence show that $$\int _ { 0 } ^ { \ln 2 } \tanh ^ { 4 } x \mathrm {~d} x = p + \ln 2$$ where \(p\) is a rational number to be found.
   8. \(\quad I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0\)
3. Show that, for \(n \geqslant 1\) $$I _ { n } = I _ { n - 1 } - \frac { 1 } { 2 n - 1 } \left( \frac { 3 } { 5 } \right) ^ { 2 n - 1 }$$

7. Given that $$I _ { n } = \int \frac { \sin n x } { \sin x } \mathrm {~d} x , \quad n \geqslant 1$$

1. prove that, for \(n \geqslant 3\) $$I _ { n } - I _ { n - 2 } = \int 2 \cos ( n - 1 ) x \mathrm {~d} x$$
2. Hence, showing each step of your working, find the exact value of $$\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 6 } } \frac { \sin 5 x } { \sin x } d x$$ giving your answer in the form \(\frac { 1 } { 12 } ( a \pi + b \sqrt { 3 } + c )\), where \(a\), \(b\) and \(c\) are integers to be found.

5 It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { n } x \mathrm {~d} x$$

1. By considering \(I _ { n } + I _ { n - 2 }\), or otherwise, show that, for \(n \geqslant 2\), $$( n - 1 ) \left( I _ { n } + I _ { n - 2 } \right) = 1 .$$
2. Find \(I _ { 4 }\) in terms of \(\pi\).

9

1. Show that \(\tanh ( \ln n ) = \frac { n ^ { 2 } - 1 } { n ^ { 2 } + 1 }\). It is given that, for non-negative integers \(n , I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { n } u \mathrm {~d} u\).
2. Show that \(I _ { n } - I _ { n - 2 } = - \frac { 1 } { n - 1 } \left( \frac { 3 } { 5 } \right) ^ { n - 1 }\), for \(n \geqslant 2\).
3. Find the value of \(I _ { 3 }\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants.
4. Use the method of differences on the result of part (ii) to find the sum of the infinite series $$\frac { 1 } { 2 } \left( \frac { 3 } { 5 } \right) ^ { 2 } + \frac { 1 } { 4 } \left( \frac { 3 } { 5 } \right) ^ { 4 } + \frac { 1 } { 6 } \left( \frac { 3 } { 5 } \right) ^ { 6 } + \ldots .$$

5 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { n } x \mathrm {~d} x$$ where \(n \geqslant 0\). Use the fact that \(\tan ^ { 2 } x = \sec ^ { 2 } x - 1\) to show that, for \(n \geqslant 2\), $$I _ { n } = \frac { 1 } { n - 1 } - I _ { n - 2 }$$ Show that \(I _ { 8 } = \frac { 1 } { 7 } - \frac { 1 } { 5 } + \frac { 1 } { 3 } - 1 + \frac { 1 } { 4 } \pi\).

10 It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 2 n } x } { \cos x } \mathrm {~d} x\), where \(n \geqslant 0\). Show that $$I _ { n } - I _ { n + 1 } = \frac { 2 ^ { - \left( n + \frac { 1 } { 2 } \right) } } { 2 n + 1 }$$ Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 6 } x } { \cos x } \mathrm {~d} x = \ln ( 1 + \sqrt { } 2 ) - \frac { 73 } { 120 } \sqrt { } 2\).

5 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 2 n \theta } { \cos \theta } \mathrm {~d} \theta\), where \(n\) is a non-negative integer.

1. Use the identity \(\sin P + \sin Q \equiv 2 \sin \frac { 1 } { 2 } ( P + Q ) \cos \frac { 1 } { 2 } ( P - Q )\) to show that $$I _ { n } + I _ { n - 1 } = \frac { 2 } { 2 n - 1 } , \text { for all positive integers } n$$
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 8 \theta } { \cos \theta } d \theta\).

3 For integers \(n \geqslant 0 , \mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } \frac { \mathrm { x } ^ { \mathrm { n } } } { 1 + \mathrm { x } ^ { 2 } } \mathrm { dx }\).

1. For integers \(n \geqslant 2\), show that \(I _ { n } + I _ { n - 2 } = \frac { 1 } { n - 1 }\).
2. 1. Determine the exact value of \(I _ { 10 }\).
   2. Deduce that \(\pi < 3 \frac { 107 } { 315 }\).

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$I _ { n } = \int \frac { \cos ( n x ) } { \sin x } \mathrm {~d} x \quad n \geqslant 1$$

1. Show that, for \(n \geqslant 1\) $$I _ { n + 2 } = \frac { 2 \cos ( n + 1 ) x } { n + 1 } + I _ { n }$$
2. Hence determine the exact value of $$\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \frac { \cos ( 5 x ) } { \sin x } d x$$ giving the answer in the form \(a + b \ln c\) where \(a , b\) and \(c\) are rational numbers to be found.