Hard +2.3 This AEA question requires sophisticated integration techniques across multiple parts, building from proving a substitution identity to applying it in non-obvious ways. Parts (b) and (d) require recognizing clever symmetry arguments, while part (c) demands algebraic manipulation followed by strategic application of earlier results. The multi-layered structure and need for creative insight place this well above standard A-level integration questions.
(a) Given that f is a function such that the integrals exist,
use the substitution \(u = a - x\) to show that
$$\int _ { 0 } ^ { a } \mathrm { f } ( x ) \mathrm { d } x = \int _ { 0 } ^ { a } \mathrm { f } ( a - x ) \mathrm { d } x$$
Hence use symmetry of \(\mathrm { f } ( \sin x )\) on the interval \([ 0 , \pi ]\) to show that
$$\int _ { 0 } ^ { \pi } x \mathrm { f } ( \sin x ) \mathrm { d } x = \pi \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { f } ( \sin x ) \mathrm { d } x$$
(b) Use the result of (a)(i) to show that
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { n } x } { \sin ^ { n } x + \cos ^ { n } x } \mathrm {~d} x$$
is independent of \(n\), and find the value of this integral.
(c) (i) Prove that
$$\frac { \cos x } { 1 + \cos x } \equiv 1 - \frac { 1 } { 2 } \sec ^ { 2 } \left( \frac { x } { 2 } \right)$$
Hence use the results from (a) to find
$$\int _ { 0 } ^ { \pi } \frac { x \sin x } { 1 + \sin x } \mathrm {~d} x$$
(d) Find
$$\int _ { 0 } ^ { \pi } \frac { x \sin ^ { 4 } x } { \sin ^ { 4 } x + \cos ^ { 4 } x } \mathrm {~d} x$$
\begin{enumerate}
\item (a) Given that f is a function such that the integrals exist,\\
(i) use the substitution $u = a - x$ to show that
\end{enumerate}
$$\int _ { 0 } ^ { a } \mathrm { f } ( x ) \mathrm { d } x = \int _ { 0 } ^ { a } \mathrm { f } ( a - x ) \mathrm { d } x$$
(ii) Hence use symmetry of $\mathrm { f } ( \sin x )$ on the interval $[ 0 , \pi ]$ to show that
$$\int _ { 0 } ^ { \pi } x \mathrm { f } ( \sin x ) \mathrm { d } x = \pi \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { f } ( \sin x ) \mathrm { d } x$$
(b) Use the result of (a)(i) to show that
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { n } x } { \sin ^ { n } x + \cos ^ { n } x } \mathrm {~d} x$$
is independent of $n$, and find the value of this integral.\\
(c) (i) Prove that
$$\frac { \cos x } { 1 + \cos x } \equiv 1 - \frac { 1 } { 2 } \sec ^ { 2 } \left( \frac { x } { 2 } \right)$$
(ii) Hence use the results from (a) to find
$$\int _ { 0 } ^ { \pi } \frac { x \sin x } { 1 + \sin x } \mathrm {~d} x$$
(d) Find
$$\int _ { 0 } ^ { \pi } \frac { x \sin ^ { 4 } x } { \sin ^ { 4 } x + \cos ^ { 4 } x } \mathrm {~d} x$$
\hfill \mbox{\textit{Edexcel AEA 2020 Q6 [23]}}