Calculus

Definite Integrals

Definite Integrals: Level 3 Challenges

         

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

Details and assumptions:

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

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

  • {x}\{x\} is the fractional part of xx.

Let Im=02πcos(x)cos(2x)cos(mx)dxI_m = \displaystyle \int_0^{2\pi} \cos(x) \cos(2x) \dots \cos(mx) dx. What is the sum of all integers 100m110100 \leq m \leq 110 such that Im0I_m \neq 0?

SN=k=1N1k\displaystyle{ S }_{ N }=\sum _{k=1}^{N}\frac1k

Let SNS_N satisfy the equation above. What is the value of limn(S2nSn)\displaystyle \lim _{ n\rightarrow \infty }{ \left( { S }_{ 2n }-{ S }_{ n } \right) }?

Give your answer to three decimal places.

f(x)={1x,x1x1,x>1 f(x) = \begin{cases}{1-|x|}, && {|x|\>\le\>1} \\ {|x|-1,} && {|x|>1}\end{cases}

g(x)=f(x1)+f(x+1)g(x) = f(x-1)+f(x+1).

Given the two functions above, what is the value of 35g(x)dx \displaystyle \int _{ -3 }^{ 5 }{ g(x) \mathrm{d}x } ?

0π/2(sin2014xcos2014x)dx=?\large \int_0^{\pi /2} ( \sin^{2014}x - \cos^{2014}x ) \,dx = \, ?

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