Find \(\int x \mathrm { e } ^ { 3 x } \mathrm {~d} x\).
Find the quotient and remainder when \(\left( x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - 9 \right)\) is divided by \(\left( x ^ { 2 } + x - 6 \right)\).
Differentiate each of the following with respect to \(x\) and simplify your answers.
\(\cot x ^ { 2 }\)
\(\frac { \sin x } { 3 + 2 \cos x }\)
(i) Expand \(( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
Hence, or otherwise, show that for small \(x\),
$$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$