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Let f(x)f(x)f(x) be a real-valued function continuous on [0,2]\left[0,2\right][0,2] such that f(x)=f(2x)f(x)=f(2x)f(x)=f(2x) for all xxx. If
∫01f(x)dx=100,\int_0^1 f(x) dx = 100,∫01f(x)dx=100,
then what is the value of
∫02f(x)dx?\int_0^2 f(x)dx ?∫02f(x)dx?
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Let a,b,ca,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.∫03(3ax2+2bx+c)dx=∫13(3ax2+2bx+c)dx. What is the value of a+b+ca+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 = \ ? ⌊10×∫2eln(x)1dx⌋= ?
Details and Assumptions:
∫012({x}2+⌊x⌋2) dx= ?\displaystyle \int_{0}^{12} \Bigl( \{x\}^2+\lfloor x \rfloor ^2\Bigr) \ dx = \ ? ∫012({x}2+⌊x⌋2) dx= ?
Details and assumptions:
If ∫03f(x) dx=2 \int_0 ^ {3} f(x) \, dx = 2 ∫03f(x)dx=2, what is the value of
∫03(3−f(x)) dx? \int_0^{3} \left( 3 - f(x) \right) \, dx ? ∫03(3−f(x))dx?
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