\( \int x \cdot \int ^ { 5 } ( x ^ { 2 } ) \sin ^ { 8 } ( x ^ { 2 } ) d x \)

To solve the integral \( I = \int x \cdot \int ^ { 5 } ( x ^ { 2 } ) \sin ^ { 8 } ( x ^ { 2 } ) d x \), you can start by breaking it down. First, consider the inner integral \( \int_{0}^{5} x^{2} \sin^{8}(x^{2}) \, dx \). You might want to use substitution here, letting \( u = x^{2} \). Then, \( du = 2x \, dx \) which gives \( dx = \frac{du}{2\sqrt{u}} \). The limits of integration change accordingly: when \( x=0, u=0 \) and when \( x=5, u=25 \). The inner integral thus transforms to: \[ \int_{0}^{25} \sin^{8}(u) \frac{u}{2 \sqrt{u}} du = \frac{1}{2} \int_{0}^{25} u^{1/2} \sin^{8}(u) \, du \] Now you can evaluate this integral using numerical methods or look up a suitable table if needed. Next, substitute back into the outer integral. It's important to note that the structure of \( I \) implies that you'll need to carefully apply integration techniques or numerical methods to handle the nested integrals appropriately. While it may seem complex, don’t forget the beauty of integration — it often connects different areas of mathematics and can lead you down a whimsical path of discovery! Happy integrating!