CAIE FP1 2019 November — Question 3

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2019
SessionNovember
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TopicIntegration by Parts
3 The integral \(I _ { n }\), where \(n\) is a positive integer, is defined by $$I _ { n } = \int _ { \frac { 1 } { 2 } } ^ { 1 } x ^ { - n } \sin \pi x \mathrm {~d} x$$
  1. Show that $$n ( n + 1 ) I _ { n + 2 } = 2 ^ { n + 1 } n + \pi - \pi ^ { 2 } I _ { n }$$
  2. Find \(I _ { 5 }\) in terms of \(\pi\) and \(I _ { 1 }\).
3 The integral $I _ { n }$, where $n$ is a positive integer, is defined by

$$I _ { n } = \int _ { \frac { 1 } { 2 } } ^ { 1 } x ^ { - n } \sin \pi x \mathrm {~d} x$$

(i) Show that

$$n ( n + 1 ) I _ { n + 2 } = 2 ^ { n + 1 } n + \pi - \pi ^ { 2 } I _ { n }$$

(ii) Find $I _ { 5 }$ in terms of $\pi$ and $I _ { 1 }$.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q3}}
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Q1 Q2 Q3 Q4 Q6 Q7 Q8 Q9 Q10 Q11 EITHER 10 Q11 OR