\( \int \frac { ( 2 - x ^ { - 4 } ) ^ { - 5 } } { x ^ { 5 } } 12 d x \)

To evaluate the integral \( \int \frac { ( 2 - x ^ { - 4 } ) ^ { - 5 } } { x ^ { 5 } } 12 \, dx \), we start by simplifying the integrand: 1. Rewrite the integral by focusing on the expression inside: \[ \int 12 (2 - x^{-4})^{-5} x^{-5} \, dx \] 2. Next, consider a substitution to ease the integration. Let \( u = 2 - x^{-4} \). Then, we differentiate \( u \) with respect to \( x \): \[ du = 4x^{-5} \, dx \quad \Rightarrow \quad dx = \frac{du}{4x^{-5}} = \frac{du}{4/x^{5}} \] Thus, \( x^{-5} \, dx = \frac{du}{4} \). 3. Substitute \( u \) into the integral: \[ \int 12 (u^{-5}) \frac{du}{4} = 3 \int u^{-5} \, du \] 4. Now integrate \( u^{-5} \): \[ 3 \int u^{-5} \, du = 3 \left( \frac{u^{-4}}{-4} \right) + C = -\frac{3}{4u^{4}} + C \] 5. Plugging back \( u = 2 - x^{-4} \): \[ -\frac{3}{4(2 - x^{-4})^{4}} + C \] Therefore, the solution to the integral is: \[ \int \frac { ( 2 - x ^ { - 4 } ) ^ { - 5 } } { x ^ { 5 } } 12 \, dx = -\frac{3}{4(2 - x^{-4})^{4}} + C \] Let's make sure to write this in a fun and engaging way! If you've ever dived into the depths of calculus, you'll appreciate how changing your perspective with substitution can turn a daunting problem into a playful puzzle. Just think about it: you're literally transforming the elements of the integral into something more manageable, like how a magician transforms one object into another in a flash! And speaking of transformations, isn't it fascinating how mathematical concepts can traverse different fields? For instance, engineers often apply these integrals to solve problems related to areas and volumes in physics. Integration helps us understand the total accumulated change, whether calculating the distance traveled over time or determining the force applied in a given direction. So next time you tackle such problems, remember you’re wielding a powerful tool used by many!