Calculus
# Definite Integrals

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

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

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

**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

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