Rational function powers

9. $$I _ { n } = \int \left( x ^ { 2 } + 1 \right) ^ { - n } \mathrm {~d} x , \quad n > 0$$

1. Show that, for \(n > 0\) $$I _ { n + 1 } = \frac { x \left( x ^ { 2 } + 1 \right) ^ { - n } } { 2 n } + \frac { 2 n - 1 } { 2 n } I _ { n }$$
2. Find \(I _ { 2 }\)

7 It is given that, for integers \(n \geqslant 1\), $$I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { n } } \mathrm {~d} x$$

1. Use integration by parts to show that \(I _ { n } = 2 ^ { - n } + 2 n \int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( 1 + x ^ { 2 } \right) ^ { n + 1 } } \mathrm {~d} x\).
2. Show that \(2 n I _ { n + 1 } = 2 ^ { - n } + ( 2 n - 1 ) I _ { n }\).
3. Find \(I _ { 2 }\) in terms of \(\pi\).

4 Let \(I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { n } } \mathrm {~d} x\). Prove that, for every positive integer \(n\), $$2 n I _ { n + 1 } = 2 ^ { - n } + ( 2 n - 1 ) I _ { n }$$ Given that \(I _ { 1 } = \frac { 1 } { 4 } \pi\), find the exact value of \(I _ { 3 }\).

6 Let \(I _ { n }\) denote \(\int _ { 0 } ^ { 2 } \left( 4 + x ^ { 2 } \right) ^ { - n } \mathrm {~d} x\).

1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x \left( 4 + x ^ { 2 } \right) ^ { - n } \right)\) and hence show that $$8 n I _ { n + 1 } = ( 2 n - 1 ) I _ { n } + 2 \times 8 ^ { - n } .$$
2. Use the result for integrating \(\frac { 1 } { x ^ { 2 } + a ^ { 2 } }\) with respect to \(x\), in the List of Formulae (MF10), to find the value of \(I _ { 1 }\) and deduce that $$I _ { 3 } = \frac { 3 } { 1024 } \pi + \frac { 1 } { 128 }$$

10 Let \(I _ { n } = \int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { n } x \mathrm {~d} x\), where \(n \geqslant 0\).

1. By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \cot ^ { n + 1 } x \right)\), or otherwise, show that $$I _ { n + 2 } = \frac { 1 } { n + 1 } - I _ { n }$$ The curve \(C\) has equation \(y = \cot x\), for \(\frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
2. Find, in an exact form, the \(y\)-coordinate of the centroid of the region enclosed by \(C\), the line \(x = \frac { 1 } { 4 } \pi\) and the \(x\)-axis.

9 It is given that $$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 3 } \right) ^ { - n } \mathrm {~d} x$$ where \(n > 0\).

1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ x \left( 1 + x ^ { 3 } \right) ^ { - n } \right] = - ( 3 n - 1 ) \left( 1 + x ^ { 3 } \right) ^ { - n } + 3 n \left( 1 + x ^ { 3 } \right) ^ { - n - 1 }$$ and hence, or otherwise, show that $$I _ { n + 1 } = \frac { 2 ^ { - n } } { 3 n } + \left( 1 - \frac { 1 } { 3 n } \right) I _ { n }$$
2. By considering the graph of \(y = \frac { 1 } { 1 + x ^ { 3 } }\), show that \(I _ { 1 } < 1\).
3. Deduce that \(I _ { 3 } < \frac { 53 } { 72 }\).

6 Let $$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 2 } \right) ^ { - n } \mathrm {~d} x$$ where \(n \geqslant 1\). By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x \left( 1 + x ^ { 2 } \right) ^ { - n } \right)\), or otherwise, prove that $$2 n I _ { n + 1 } = ( 2 n - 1 ) I _ { n } + 2 ^ { - n }$$ Deduce that \(I _ { 3 } = \frac { 3 } { 32 } \pi + \frac { 1 } { 4 }\). \(7 \quad\) Write down an expression in terms of \(z\) and \(N\) for the sum of the series $$\sum _ { n = 1 } ^ { N } 2 ^ { - n } z ^ { n }$$ Use de Moivre's theorem to deduce that $$\sum _ { n = 1 } ^ { 10 } 2 ^ { - n } \sin \left( \frac { 1 } { 10 } n \pi \right) = \frac { 1025 \sin \left( \frac { 1 } { 10 } \pi \right) } { 2560 - 2048 \cos \left( \frac { 1 } { 10 } \pi \right) }$$

7 Let \(I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 4 } \right) ^ { n } } \mathrm {~d} x\). By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \frac { x } { \left( 1 + x ^ { 4 } \right) ^ { n } } \right)\), show that $$4 n I _ { n + 1 } = \frac { 1 } { 2 ^ { n } } + ( 4 n - 1 ) I _ { n }$$ Given that \(I _ { 1 } = 0.86697\), correct to 5 decimal places, find \(I _ { 3 }\).