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Definite Integrals

The definite integral of a function computes the area under the graph of its curve, allowing us to calculate areas and volumes that are not easily done using geometry alone.

Level 3

         

\[ \displaystyle \int_{0}^{20} \Bigl(\lfloor x \rfloor \{x\} \Bigr) \ dx = \ ? \]

Details and assumptions:

  • Every \(x\in \mathbb{R}\) can be written as \(x=\lfloor x \rfloor + \{x\} \).

  • \(\lfloor x \rfloor\) denotes greatest integer less than or equal to \(x\).

  • \(\{x\} \) is the fractional part of \(x\).

Let \(I_m = \displaystyle \int_0^{2\pi} \cos(x) \cos(2x) \dots \cos(mx) dx\). What is the sum of all integers \(100 \leq m \leq 110\) such that \(I_m \neq 0\)?

\[\displaystyle{ S }_{ N }=\sum _{k=1}^{N}\frac1k \]

Let \(S_N\) satisfy the equation above. What is the value of \(\displaystyle \lim _{ n\rightarrow \infty }{ \left( { S }_{ 2n }-{ S }_{ n } \right) }\)?

Give your answer to three decimal places.

\( f(x) = \begin{cases}{1-|x|}, && {|x|\>\le\>1} \\ {|x|-1,} && {|x|>1}\end{cases} \)

\(g(x) = f(x-1)+f(x+1)\).

Given the two functions above, what is the value of \( \displaystyle \int _{ -3 }^{ 5 }{ g(x) \mathrm{d}x } \)?

\[\large \int_0^{\pi /2} ( \sin^{2014}x - \cos^{2014}x ) \,dx = \, ? \]

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