\( \left. \begin{array} { l } { \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \sin x ) ^ { \frac { 8 } { 3 } } ( \sec x ) ^ { \frac { 1 } { 2 } } d x } \\ { \pi } \\ { \int _ { 0 } ^ { \pi } \sin x ) ^ { \frac { 7 } { 2 } } \cos ^ { \frac { 3 } { 2 } } d x } \\ { \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \operatorname { cosec } \theta ) ^ { - \frac { 1 } { 2 } } ( \cos - \theta ^ { \frac { 4 } { 3 } } ) d \theta } \end{array} \right. \)

It seems that the integrals provided cannot be calculated directly using the tools available. However, I can guide you through the process of solving each integral step by step. ### 1. Integral \( \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \sin x ) ^ { \frac { 8 } { 3 } } ( \sec x ) ^ { \frac { 1 } { 2 } } d x \) **Step 1:** Rewrite the integral using the identity \( \sec x = \frac{1}{\cos x} \): \[ \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \sin x ) ^ { \frac { 8 } { 3 } } ( \sec x ) ^ { \frac { 1 } { 2 } } d x = \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \sin x ) ^ { \frac { 8 } { 3 } } ( \frac{1}{\cos x})^{\frac{1}{2}} d x \] **Step 2:** Simplify the integral: \[ = \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \sin x ) ^ { \frac { 8 } { 3 } } ( \cos x )^{- \frac{1}{2}} d x \] **Step 3:** Use the substitution \( u = \sin x \), \( du = \cos x \, dx \): \[ = \int _ { 0 } ^ { 1 } u^{\frac{8}{3}} (1-u^2)^{- \frac{1}{2}} du \] This integral can be evaluated using the Beta function or by recognizing it as a form of the Gamma function. ### 2. Integral \( \int _ { 0 } ^ { \pi } ( \sin x ) ^ { \frac { 7 } { 2 } } \cos ^ { \frac { 3 } { 2 } } d x \) **Step 1:** Use the identity \( \cos^{\frac{3}{2}} x = (\cos x)^{\frac{3}{2}} \): \[ \int _ { 0 } ^ { \pi } ( \sin x ) ^ { \frac { 7 } { 2 } } ( \cos x )^{\frac{3}{2}} d x \] **Step 2:** This integral can be evaluated using the Beta function: \[ = 2 \int _ { 0 } ^ { \frac{\pi}{2}} ( \sin x )^{\frac{7}{2}} ( \cos x )^{\frac{3}{2}} d x \] ### 3. Integral \( \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \operatorname { cosec } \theta ) ^ { - \frac { 1 } { 2 } } ( \cos - \theta ^ { \frac { 4 } { 3 } } ) d \theta \) **Step 1:** Rewrite using \( \csc \theta = \frac{1}{\sin \theta} \): \[ \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \sin \theta )^{\frac{1}{2}} ( \cos(-\theta))^{\frac{4}{3}} d \theta \] **Step 2:** Since \( \cos(-\theta) = \cos(\theta) \): \[ = \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \sin \theta )^{\frac{1}{2}} ( \cos \theta)^{\frac{4}{3}} d \theta \] This integral can also be evaluated using the Beta function. ### Conclusion To evaluate these integrals, you can use the properties of the Beta and Gamma functions. If you need further assistance with specific calculations or numerical evaluations, please let me know!