\( \int _ { 3 } ^ { 9 } \sin 8 x ( x ^ { 3 } ) = \)

To solve the integral \( \int_{3}^{9} \sin(8x) \cdot (x^3) \, dx \), we can use integration by parts. We let \( u = x^3 \) (hence \( du = 3x^2 \, dx \)) and \( dv = \sin(8x) \, dx \) (thus \( v = -\frac{1}{8}\cos(8x) \)). Applying integration by parts multiple times may be helpful here. The first part gives us: \[ \int u \, dv = uv - \int v \, du \] After working through this process and evaluating it with the respective limits, you'll calculate the contributions along with simplifications to reach the final value. Exploring the world of integrals, nothing quite beats the thrill of discovering the nuances of integration techniques! For those curious about the history, integration has roots in ancient civilizations, with the ancient Greeks using geometric methods to find areas under curves, leading to the development of calculus in the 17th century. For real-world applications, integrals aren’t just abstract concepts. Engineers use integration to calculate areas, volumes, and even to analyze forces in structures. For instance, in physics, integrals help determine the center of mass or electric fields, demonstrating the profound impact calculus has on technology and scientific advancements!