Challenging +1.8 This is a substantial Further Maths question requiring integration by parts twice (part i), deriving a reduction formula through integration by parts and trigonometric identities (part ii), and applying the formula to find a centroid (part iii). While technically demanding with multiple steps, the techniques are standard for FP1 reduction formula questions. The symmetric limits simplify some work, and the question provides clear scaffolding through its parts.
It is given that, for \(n \geqslant 0\),
$$I _ { n } = \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos ^ { n } x \mathrm {~d} x$$
Show that, for \(n \geqslant 2\),
$$4 I _ { n } = n ( n - 1 ) \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin ^ { 2 } x \cos ^ { n - 2 } x \mathrm {~d} x - n I _ { n }$$
and deduce the reduction formula
$$\left( n ^ { 2 } + 4 \right) I _ { n } = n ( n - 1 ) I _ { n - 2 }$$
Using the result in part (i) and the reduction formula in part (ii), find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis and the arc of the curve \(y = \mathrm { e } ^ { x } \cos x\) from \(x = - \frac { 1 } { 2 } \pi\) to \(x = \frac { 1 } { 2 } \pi\). Give your answer correct to 3 significant figures.
(i) Show that
$$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { x } \cos x \mathrm {~d} x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \frac { 1 } { 2 } \pi } + \mathrm { e } ^ { - \frac { 1 } { 2 } \pi } \right)$$
(ii) It is given that, for $n \geqslant 0$,
$$I _ { n } = \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos ^ { n } x \mathrm {~d} x$$
Show that, for $n \geqslant 2$,
$$4 I _ { n } = n ( n - 1 ) \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin ^ { 2 } x \cos ^ { n - 2 } x \mathrm {~d} x - n I _ { n }$$
and deduce the reduction formula
$$\left( n ^ { 2 } + 4 \right) I _ { n } = n ( n - 1 ) I _ { n - 2 }$$
(iii) Using the result in part (i) and the reduction formula in part (ii), find the $y$-coordinate of the centroid of the region bounded by the $x$-axis and the arc of the curve $y = \mathrm { e } ^ { x } \cos x$ from $x = - \frac { 1 } { 2 } \pi$ to $x = \frac { 1 } { 2 } \pi$. Give your answer correct to 3 significant figures.\\
\hfill \mbox{\textit{CAIE FP1 2018 Q11 EITHER}}