Finding $\int\lfloor x\rfloor\cdot |\sin(\pi x)| dx$

On each interval $[2k,2k+1)$, because $\sin(\pi x)$ is non-negative, the integral is equal to

$$f(x) = \lfloor x\rfloor\int |\sin(\pi x)| dx = \lfloor x\rfloor\int \sin(\pi x) dx$$

Solving this integral, you get

$$f(x) = f_1(x) = -\frac{\lfloor x\rfloor}{\pi}\cos(\pi x) + A(\lfloor x\rfloor)$$

On the intervals $[2k-1, 2k)$, you will have the same result, just with a different sign,

$$f(x) = f_2(x) = \frac{\lfloor x\rfloor}{\pi}\cos(\pi x) + B(\lfloor x\rfloor)$$

Notice that the above "constants" of integration are functions of $\lfloor x\rfloor$. This is because we have integrated separately on each interval, and therefore the constants for each interval are different. To determine these constants, we will use the fact that $f(x)$ is differentiable and therefore continuous.

By continuity at the points $x=2k$, we have

\begin{eqnarray*} &f_1(2k)=\lim\limits_{x\to2k^-} f_2(x)& \\ &\frac{2k}{\pi}\cos(2k\pi) + A(2k) = \frac{2k-1}{\pi}\cos(2k\pi) + B(2k-1)& \\ &\boxed{B(2k-1) = A(2k) + \frac{1-4k}{\pi}} \tag{1}& \end{eqnarray*}

By continuity at the points $x=2k+1$, we have

\begin{eqnarray*} &\lim\limits_{x\to(2k+1)^-} f_1(x) = f_2(2k+1)& \\ &-\frac{2k}{\pi}\cos(2k+1)\pi + A(2k) = \frac{2k+1}{\pi}\cos(2k+1)\pi + B(2k+1)& \\ &\boxed{A(2k) = -\frac{4k+1}{\pi} + B(2k+1)} \tag{2}& \end{eqnarray*}

Next, since the integrand is identically zero on $[0,1)$, $f(x)=C$ is constant on this interval. Because $[0,1)$ is of the form $[2k,2k+1)$ for $k=0$, this gives us $A(0) = C$. Consequently, $(2)$ yields $B(1) = \frac{1}{\pi}+C$.

Combining $(1)$ and $(2)$ we get

\begin{eqnarray*} &B(2k+1) = \frac{8k}{\pi} + B(2k-1) = \frac{8k}{\pi} + \frac{8(k-1)}{\pi} + B(2k-3) = \cdots =& \\ &= \frac{8}{\pi} (k+k-1+...+1) + B(1) = \frac{8}{\pi}\frac{k(k+1)}{2} + \frac{1}{\pi}+C =& \\ &= \frac{1}{\pi}(4k^2+4k+1)+C = \frac{1}{\pi}(2k+1)^2+C& \end{eqnarray*}

Now, applying $(2)$ we get $A(2k) = \dfrac{4k^2}{\pi}+C$. A more useful way to express the last two results is

$$B(\lfloor x\rfloor) = \frac{\lfloor x\rfloor^2}{\pi}+C,\ A(\lfloor x\rfloor) = \frac{\lfloor x\rfloor^2}{\pi}+C$$

on their respective domains.

Finally, just merge the functions $f_1$ and $f_2$ into one and get

$$\boxed{\boxed{f(x) = (-1)^{\lfloor x\rfloor+1}\cdot\frac{\lfloor x\rfloor}{\pi}\cos(\pi x) + \frac{\lfloor x\rfloor^2}{\pi}+C}}$$

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An easier way

First, notice that the integrand is a continuous function on $\mathbb R$. This allows us to use the Fundamental theorem of calculus, i.e.

$$f(x)=\int\lfloor x\rfloor\cdot |\sin(\pi x)| dx = \int_0^x\lfloor t\rfloor\cdot |\sin(\pi t)| dx$$

We can forget about the constant of integration for now.

Next, for $x\ge 1$, we can write

$$f(x) = \sum_{k=1}^{\lfloor x\rfloor} \int_{k-1}^k \lfloor t\rfloor\cdot |\sin(\pi t)| dt + \int_{\lfloor x\rfloor}^x\lfloor t\rfloor\cdot |\sin(\pi t)|dt$$

Use the fact that $\lfloor t \rfloor = k-1$ on the interval $[k-1,k)$, and you will get $$\int_{k-1}^k \lfloor t\rfloor\cdot |\sin(\pi t)| dt = (k-1)\int_{k-1}^k |\sin(\pi t)| dt$$

Because $|\sin(\pi t)|$ is $1-$periodic, the integration interval $[k-1, k]$ can be reduced to the interval $[0, 1]$, which makes it easy to calculate it (the result is $2/\pi$).

Another thing you will need to use is the fact that

$$\int_{\lfloor x\rfloor}^x\lfloor t\rfloor\cdot |\sin(\pi t)|dt = \lfloor x\rfloor \int_{\lfloor x\rfloor}^x |\sin(\pi t)|dt = \lfloor x\rfloor \int_0^{x-\lfloor x\rfloor} \sin(\pi t)dt$$

For $x<1$, it can be shown that the result is the same. Also, you can add a constant of integration if you like.

In every unit interval $[k,k+1[$, the first factor is the constant $k$, while the second is a positive arch of sinusoid.

For $k=0$, we have

$$\int_0^x\sin(\pi x)\,dx=-\left.\frac{\cos(\pi x)}{\pi}\right|_0^x=\frac{1-\cos(\pi x)}{\pi},$$ which runs from $0$ to $\dfrac2\pi$. For the next intervals, the expression is the same with amplitude multiplied by $k$, and with an additional term $\dfrac{k(k-1)}{\pi}$ for continuity (triangular number).

Hence after simplification

$$\lfloor x\rfloor\frac{\lfloor x\rfloor-\cos(\pi\{x\})}{\pi}.$$

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