If  bold italic f bold left parenthesis bold italic x bold right parenthesis bold space bold equals bold space fraction numerator bold g bold left parenthesis bold x bold right parenthesis over denominator bold h bold left parenthesis bold x bold right parenthesis end fraction where  g(x) = 60000 x2 + 600sin x and h(x) = (  2 x3 + 7  )3,

then  bold integral subscript bold minus bold 1 end subscript superscript bold 1 bold italic f bold left parenthesis bold italic x bold right parenthesis bold space bold d bold italic x bold space bold equals

We are tasked with calculating the integral of \( f(x) \), where:

\[
f(x) = \frac{g(x)}{h(x)} = \frac{60000x^2 + 600 \sin x}{(2x^3 + 7)^3}
\]

We need to compute the definite integral:

\[
\int_{-1}^{1} f(x) \, dx = \int_{-1}^{1} \frac{60000x^2 + 600 \sin x}{(2x^3 + 7)^3} \, dx
\]

### Approach

1. **Simplification and symmetry**: We examine the integrand to check if it's even, odd, or neither. If the function is odd, the integral from \(-1\) to \(1\) would be zero, simplifying the process.

- \( g(x) = 60000x^2 + 600 \sin x \): 
  - The term \( 60000x^2 \) is **even** (since \( x^2 \) is even).
  - The term \( 600 \sin x \) is **odd** (since \( \sin x \) is odd).
  
  Thus, \( g(x) \) is neither even nor odd, so no simplification can be made here.

- \( h(x) = (2x^3 + 7)^3 \):
  - The term \( 2x^3 \) is odd, but the constant 7 makes the entire expression inside the cube neither even nor odd.
  
  Hence, \( h(x) \) is neither even nor odd.

2. **Numerical integration**: Given the complexity of the integrand, it is unlikely that a straightforward analytical solution is available. Thus, we will proceed with a numerical evaluation of the definite integral.

Let me calculate the value of this integral numerically.The value of the definite integral \( \int_{-1}^{1} f(x) \, dx \), where \( f(x) = \frac{60000x^2 + 600 \sin x}{(2x^3 + 7)^3} \), is approximately:

\[
\int_{-1}^{1} f(x) \, dx \approx 137.67
\]

This result was computed numerically with a negligible error margin.

Would you like further details or have any questions?

Here are five related questions to expand on this concept:
1. How would the integral change if \( h(x) \) had a different power, say \( (2x^3 + 7)^2 \)?
2. What techniques can be used for approximating integrals when no analytical solution exists?
3. How does the symmetry of a function affect the evaluation of definite integrals?
4. Can this type of integral be solved using methods like substitution or partial fractions?
5. How does the behavior of \( f(x) \) change for larger intervals, such as \( \int_{-2}^{2} f(x) \, dx \)?

**Tip:** Always check for symmetry when calculating definite integrals, as it can sometimes simplify the evaluation significantly.

If f(x) = g(x)/h(x) where g(x) = 60000x^2 + 600sin(x) and h(x) = (2x^3 + 7)^3, then ∫ from -1 to 1 of f(x) dx equals?

Discover how to evaluate the definite integral of f(x) = (60000x^2 + 600sin(x)) / (2x^3 + 7)^3 from -1 to 1. This problem explores concepts like numerical integration, symmetry, and function behavior, making it suitable for advanced high school or early college students.

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