Derive reduction formula by differentiation

6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { 1 }\).
2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined. \includegraphics[max width=\textwidth, alt={}, center]{20e14db3-0eb0-4954-91cf-027e16f8bf14-11_78_1576_336_321}

6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { 1 }\).
2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined. \includegraphics[max width=\textwidth, alt={}, center]{1de67949-6262-4ade-b986-02b6563ae404-11_78_1576_336_321}

7 The integral \(\mathrm { I } _ { \mathrm { n } }\), where n is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 3 } { 2 } } \left( 4 + \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { 1 }\), expressing your answer in logarithmic form.
2. By considering \(\frac { d } { d x } \left( x \left( 4 + x ^ { 2 } \right) ^ { - \frac { 1 } { 2 } n } \right)\), or otherwise, show that $$4 n l _ { n + 2 } = \frac { 3 } { 2 } \left( \frac { 2 } { 5 } \right) ^ { n } + ( n - 1 ) l _ { n } .$$
3. Find the value of \(I _ { 5 }\).

4 The integral \(\mathrm { I } _ { \mathrm { n } }\) is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } \left( 1 + \mathrm { x } ^ { 5 } \right) ^ { \mathrm { n } } \mathrm { dx }\).

1. By considering \(\frac { d } { d x } \left( x \left( 1 + x ^ { 5 } \right) ^ { n } \right)\), or otherwise, show that $$( 5 n + 1 ) l _ { n } = 2 ^ { n } + 5 n l _ { n - 1 }$$
2. Find the exact value of \(I _ { 3 }\).

7 The integral \(\mathrm { I } _ { \mathrm { n } }\), where n is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 4 } { 3 } } \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { - 1 }\) giving your answer in the form \(\ln a\), where \(a\) is an integer to be determined.
2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm { n } \right)\), or otherwise, show that $$( \mathrm { n } + 1 ) \mathrm { I } _ { \mathrm { n } } = \mathrm { nl } _ { \mathrm { n } - 2 } + \frac { 4 } { 3 } \left( \frac { 5 } { 3 } \right) ^ { \mathrm { n } }$$
3. A curve has equation \(y = x ^ { 2 }\), for \(0 \leqslant x \leqslant \frac { 2 } { 3 }\). The arc length of the curve is denoted by \(s\). Use the substitution \(\mathrm { u } = 2 \mathrm { x }\) to show that \(\mathrm { s } = \frac { 1 } { 2 } \mathrm { l } _ { 1 }\) and find the exact value of \(s\).

2 The integral \(I _ { n }\), where \(n\) is a non-negative integer, is defined by $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x ^ { 3 } } \mathrm {~d} x$$ By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { n + 1 } \mathrm { e } ^ { - x ^ { 3 } } \right)\) or otherwise, show that $$3 I _ { n + 3 } = ( n + 1 ) I _ { n } - \mathrm { e } ^ { - 1 }$$ Hence find \(I _ { 6 }\) in terms of e and \(I _ { 0 }\).

8 Let \(I _ { n } = \int _ { 0 } ^ { \ln 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n } \mathrm {~d} x\).

1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right) \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n - 1 } \right] = n \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n } - 4 ( n - 1 ) \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n - 2 } .$$
2. Hence show that $$n I _ { n } = 4 ( n - 1 ) I _ { n - 2 } + \frac { 3 } { 2 } \left( \frac { 5 } { 2 } \right) ^ { n - 1 } .$$
3. Use the result in part (ii) to find the \(y\)-coordinate of the centroid of the region bounded by the axes, the line \(x = \ln 2\) and the curve $$y = \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 2 } .$$ Give your answer correct to 3 decimal places.