Methods in Calculus (Edexcel A Level Further Maths: Further Pure 1): Exam Questions

$y={\text{e}}^{2x}\text{sin}x$

Use Leibnitz's theorem to show that

$\frac{{\text{d}}^{4}y}{\text{d}{x}^4}={\text{e}}^{2x}(24\text{cos}x-7\text{sin}x)$

Hence express $\frac{{\text{d}}^{4}y}{\text{d}{x}^4}$ in the form $R{\text{e}}^{2x}\text{cos}(x+\alpha )$, where $R$ and $\alpha$ are constants to be determined, $R>0$ and $0<\alpha <\frac{\pi }{2}$.

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Use the substitution $t=\text{tan}\frac{\theta }{2}$ to show that

$\int \frac{1}{2+\text{sin}\theta }\text{d}\theta =\int \frac{a}{t^2+bt+c}\text{d}t$

where $a$, $b$ and $c$ are constants to be determined.

Hence show that

${\int }_0^{\frac{\pi }{2}}\frac{1}{2+\text{sin}\theta }\text{d}\theta =\frac{\sqrt{3}\pi }{9}$