$\int_{0}^{1} \frac{ \lfloor2x\rfloor\cdot x}{ \lfloor2x\rfloor+x^2} \ dx $

Question

I need help to know if I worked the example correctly or incorrectly

Answer 1

$$\int_0^1 \frac{x\lfloor 2 x\rfloor }{x^2+ \lfloor 2 x\rfloor } \, dx=\int_{\frac{1}{2}}^1 \frac{x}{x^2+1} \, dx=\frac{1}{2} \log \left(\frac{8}{5}\right)$$

Because the floor of $2x$ is $0$ between $0$ and $\frac{1}{2}$ and $1$ between $\frac{1}{2}$ and $1$.

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