For Harper

(a) Berechne: \( \sqrt{19} \times \sqrt{2} \)

(b) Vereinfache den Ausdruck: \( \frac{\sqrt{9}}{\sqrt{10}} + \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}} \left(1 + \frac{1}{2} \sin^2(3x) + \frac{\alpha}{\beta}\sqrt{\gamma+\delta} \cdot \frac{\sqrt[3]{\theta^2 + \phi^2}}{\psi + \frac{\omega}{\chi}}\right) \,dx \)

(c) Multipliziere: \( (\sqrt{11} + \sqrt{2}) \times (\sqrt{6} - \sqrt{3}) \)

(d) Dividiere: \( \frac{\sqrt{64}}{\sqrt{6}} \)

(e) Vereinfache den Ausdruck: \( \begin{equation} x = a_0 + \frac{1}{\displaystyle a_1 + \frac{1}{\displaystyle a_2 + \frac{1}{\displaystyle a_3 + a_4}}} \end{equation} \)

(f) Berechne: \( \frac{\sqrt{100} \times \sqrt{56}}{\sqrt{12}} \)

(g) Multipliziere und vereinfache: \( (\sqrt{10} + \sqrt{6})^2 \)

(h) Dividiere: \( \frac{\sqrt{198}}{\sqrt{30}} - \begin{bmatrix} \int_{0}^{1} e^{2t}\sin(t)\,dt & \frac{\cos^2(3\theta)}{\sqrt{2}} & 0 & \frac{\pi}{4} & \ln(2) \\ \frac{\sqrt{5}}{\phi} & \int_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2}\,dx & \frac{1}{\sqrt[3]{\alpha + \beta}} & \frac{\gamma^2}{\delta} & \frac{\sqrt{\pi}}{2} \\ \frac{\theta}{2\pi} & \frac{1}{\sqrt{3}} & \int_{0}^{1} \frac{e^{2t}}{\sqrt{t}}\,dt & \frac{\omega}{\chi} & \frac{\sqrt{2}}{\sqrt{\pi}} \\ \ln(\sqrt{\pi}) & \frac{\sqrt[4]{\pi^3}}{\sqrt{\alpha}} & \frac{\sqrt{\beta}}{\sqrt[3]{\gamma}} & \int_{-\pi}^{\pi} \cos^2(\phi)\,d\phi & \frac{\sqrt[5]{\delta}}{\sqrt[6]{\varepsilon}} \end{bmatrix} \)

(i) Berechne: \( \sqrt{60} \div \sqrt{8} \)

(j) \( \mathcal{L}_{\mathcal{T}}(\vec{\lambda}) = \sum_{\mathbf{x},\mathbf{s}\in\mathcal{T}} \log P(\mathbf{x}|\mathbf{S}) - \sum_{i=1}^m \frac{\lambda_i^2}{2\sigma^2} \)