Příklad 4.48

\[ \begin{aligned} &\int_{0}^{1} \frac{2-3\arcsin(x)}{\sqrt{1-x^2}}\;dx \\ &= \int_{0}^{1} \frac{2}{\sqrt{1-x^2}}\;dx - \int_{0}^{1} \frac{3\arcsin(x)}{\sqrt{1-x^2}}\;dx \\ &= 2\int_{0}^{1} \frac{1}{\sqrt{1-x^2}}\;dx - 3\int_{0}^{1} \frac{\arcsin(x)}{\sqrt{1-x^2}}\;dx \end{aligned} \]

\[ \begin{aligned} &= 2\int_{0}^{1} \frac{1}{\sqrt{1-x^2}}\;dx - 3\int_{0}^{1} \frac{\arcsin(x)}{\sqrt{1-x^2}}\;dx \\ &= 2\int_{0}^{\frac{\pi}{2}}\;du - 3\int_{0}^{\frac{\pi}{2}} u\;du \\ &= 2\cdot\left[u\right]_{0}^{\frac{\pi}{2}} - 3\cdot\left[\frac{u^2}{2}\right]_{0}^{\frac{\pi}{2}} \\ &= \cancel{2}\frac{\pi}{\cancel{2}} - 3\cdot\left[\frac{\left(\frac{\pi}{2}\right)^2}{2} - \frac{0}{2}\right] \\ &= \pi -3 \cdot \left[\frac{\pi^2}{4}\cdot\frac{1}{2}\right] \\ &= \boxed{\pi - \frac{3\pi^2}{8}} \\ \end{aligned} \]