Challenging +1.2 This is a standard FP3 technique using complex exponentials to evaluate integrals involving e^x cos x and e^x sin x. While it requires understanding Euler's formula and complex number manipulation, the method is directly taught and the question provides the key hint about integrating complex exponentials. The algebra is straightforward once the setup is recognized, making this a typical exam question testing a specific syllabus technique rather than requiring novel insight.
The integrals \(C\) and \(S\) are defined by
$$C = \int_0^{2\pi} e^{3x} \cos 3x \, dx \quad \text{and} \quad S = \int_0^{2\pi} e^{3x} \sin 3x \, dx.$$
By considering \(C + iS\) as a single integral, show that
$$C = \frac{1}{13}(2 + 3e^\pi),$$
and obtain a similar expression for \(S\). [8]
(You may assume that the standard result for \(\int e^{kx} dx\) remains true when \(k\) is a complex constant, so that \(\int e^{(a+ib)x} dx = \frac{1}{a+ib} e^{(a+ib)x}\).)
The integrals $C$ and $S$ are defined by
$$C = \int_0^{2\pi} e^{3x} \cos 3x \, dx \quad \text{and} \quad S = \int_0^{2\pi} e^{3x} \sin 3x \, dx.$$
By considering $C + iS$ as a single integral, show that
$$C = \frac{1}{13}(2 + 3e^\pi),$$
and obtain a similar expression for $S$. [8]
(You may assume that the standard result for $\int e^{kx} dx$ remains true when $k$ is a complex constant, so that $\int e^{(a+ib)x} dx = \frac{1}{a+ib} e^{(a+ib)x}$.)
\hfill \mbox{\textit{OCR FP3 Q4 [8]}}