Challenging +1.2 Part (a) is a standard integration by substitution where recognizing u = 1 + 2cos(x) leads directly to the answer - this is routine A-level technique. Part (b) requires identifying and executing a trigonometric substitution (x = 2sin(θ)) for an algebraic integrand, which is more sophisticated but still a well-practiced Further Maths technique with straightforward execution once the substitution is chosen. The question tests competent application of standard methods rather than novel problem-solving.
10.
a) Show that
$$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { 4 \sin x } { 1 + 2 \cos x } \mathrm {~d} x = \ln a$$
where \(a\) is a rational constant to be found.
b) By using a suitable substitution, find the exact value of
$$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$
10.\\
a) Show that
$$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { 4 \sin x } { 1 + 2 \cos x } \mathrm {~d} x = \ln a$$
where $a$ is a rational constant to be found.\\
b) By using a suitable substitution, find the exact value of
$$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q10 [8]}}