Integral 21

$\begin{aligned} V & = \int_0^\pi \frac{1- \color{#3D99F6}{\cos x}}{x} dx \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \small \color{#3D99F6}{\text{By Maclaurin series}} \\ & = \int_0^\pi \frac{1- \color{#3D99F6}{\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}-...\right)}}{x} dx \\ & = \int_0^\pi \sum_{k=1}^\infty \frac{(-1)^{k+1}x^{2k-1}}{(2k)!} dx \\ & = \sum_{k=1}^\infty \frac{(-1)^{k+1}\pi^{2k}}{2k(2k)!} \end{aligned}$

Since cosine integral $\displaystyle \ce{Ci}(x) = \gamma + \ln x + \sum_{k=1}^\infty \frac{\left(-x^2\right)^k}{2k(2k)!}$ , where $\gamma$ is Euler-Mascheroni constant , then we have:

$\begin{aligned} V & = - \sum_{k=1}^\infty \frac{\left(-\pi^2\right)k}{2k(2k)!} \\ & = \gamma + \ln \pi - \ce{Ci}(\pi) \\ & = 0.57721566 + 1.144729886 - 0.073667912 \\ & = 1.648277634 \end{aligned}$

$\Rightarrow \lfloor 10^5V \rfloor = \boxed{164827}$