Definite Integrals

Definite Integrals: Level 2 Challenges


Let f(x)f(x) be a real-valued function continuous on [0,2]\left[0,2\right] such that f(x)=f(2x)f(x)=f(2x) for all xx. If

01f(x)dx=100,\int_0^1 f(x) dx = 100,

then what is the value of

02f(x)dx?\int_0^2 f(x)dx ?

Let a,b,ca,b,c be non-zero real constants such that 03(3ax2+2bx+c)dx=13(3ax2+2bx+c)dx.\displaystyle \int_{0}^3 (3ax^2+2bx+c) \, dx=\displaystyle \int_{1}^3 (3ax^2+2bx+c) \, dx. What is the value of a+b+ca+b+c?

  10×2e ⁣1ln(x)dx  = ?\bigg \lfloor \;10 \times \displaystyle \int_2^e \! \frac {1}{\ln(x)} \, \mathrm{d}x\;\bigg \rfloor = \ ?

Details and Assumptions:

  • You may use the following approximations: e2.718e \approx 2.718, and ln(2)0.693\ln(2) \approx 0.693, and use the following graph of f(x)=1ln(x)f(x)= \frac {1}{\ln(x)}.

012({x}2+x2) dx= ?\displaystyle \int_{0}^{12} \Bigl( \{x\}^2+\lfloor x \rfloor ^2\Bigr) \ dx = \ ?

Details and assumptions:

  • Every xRx\in \mathbb{R} can be written as x=x+{x}x=\lfloor x \rfloor + \{x\} .
  • As usual, x\lfloor x \rfloor denotes greatest integer less than or equal to xx.
  • {x}\{x\} is the fractional part of xx.

If 03f(x)dx=2 \int_0 ^ {3} f(x) \, dx = 2 , what is the value of

03(3f(x))dx? \int_0^{3} \left( 3 - f(x) \right) \, dx ?


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