These Ten-Year-Series (TYS) worked solutions with video explanations for 2023 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
(a)
Find $\int{\cos px\,\,\cos \,\,qx\,\text{d}x}$, where $p$ and $q$ are constants such that $p\ne q$ and $p\ne -q$.
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(a) Find $\int{\cos px\,\,\cos \,\,qx\,\text{d}x}$, where $p$ and $q$ are constants such that $p\ne q$ and $p\ne -q$.
[2]
(b)
Given that $n\ne 0$, show that $\int{x\cos nx\,\,\text{d}x}=\frac{x\sin nx}{n}+\frac{\cos nx}{{{n}^{2}}}+c$, where $c$ is an arbitrary constant.
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(b) Given that $n\ne 0$, show that $\int{x\cos nx\,\,\text{d}x}=\frac{x\sin nx}{n}+\frac{\cos nx}{{{n}^{2}}}+c$, where $c$ is an arbitrary constant.
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(c)
Using the result in part (b) show that, for all positive integers $n$, the value of $\int_{0}^{\pi }{x\cos nx\,\text{d}x}$ can be expressed as $\frac{k}{{{n}^{2}}}$, where the possible value(s) of are to be determined.
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(c) Using the result in part (b) show that, for all positive integers $n$, the value of $\int_{0}^{\pi }{x\cos nx\,\text{d}x}$ can be expressed as $\frac{k}{{{n}^{2}}}$, where the possible value(s) of are to be determined.
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(d)
Using the result in part (b) find the exact value of $\int_{0}^{\frac{\pi }{2}}{\left| x\cos 2x \right|\text{d}x}$.
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(d) Using the result in part (b) find the exact value of $\int_{0}^{\frac{\pi }{2}}{\left| x\cos 2x \right|\text{d}x}$.
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