Definite Integrals: Level 2 Challenges Practice Problems Online

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

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

then what is the value of

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

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

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

Details and Assumptions:

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

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

Details and assumptions:

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

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

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

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Definite Integrals: Level 2 Challenges