Write down the derivative of ${{\sin }^{n}}\theta $ with respect to $\theta $, where $n$

Question

Write down the derivative of ${{\sin }^{n}}\theta $ with respect to $\theta $, where $n$ is a constant and $n\ge 1$. [1] Using the substitution $x=2{{\cos }^{2}}\theta $ for $0\le \theta \le \frac{\pi }{2}$, show that $\displaystyle\int{x}\sqrt{1-\frac{x}{2}}\ \mathrm{d}x=8\displaystyle\int{\left( \cos \theta {{\sin }^{4}}\theta -\cos \theta {{\sin }^{2}}\theta \right)}\ \mathrm{d}\theta $. [4] Hence find $\displaystyle\int{x}\sqrt{1-\frac{x}{2}}\ \mathrm{d}x$. [3]

Accepted Answer

(a) $n\sin^{n-1}\theta\cos\theta$. (b) Shown. (c) $\frac85(1-\frac x2)^{\frac52}-\frac83(1-\frac x2)^{\frac32}+C$.

2024 EJC Promo Q6

2024 EJC Promo Q6

Solution:

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