integral from 1 to 6 of sqrt(9/x^2 +x^2/133 +1/2) dx

We are asked to compute the integral:

\[
\int_1^6 \sqrt{\frac{9}{x^2} + \frac{x^2}{133} + \frac{1}{2}} \, dx
\]

This integral involves a combination of rational functions and radicals, so we'll break it down step by step.

### Step 1: Simplify the integrand
The integrand is:

\[
\sqrt{\frac{9}{x^2} + \frac{x^2}{133} + \frac{1}{2}}
\]

We'll examine each term:

1. \(\frac{9}{x^2} = 9x^{-2}\),
2. \(\frac{x^2}{133}\) is already simplified,
3. \(\frac{1}{2}\) is a constant.

Thus, the integrand becomes:

\[
\sqrt{9x^{-2} + \frac{x^2}{133} + \frac{1}{2}}.
\]

### Step 2: Evaluate the integral

Since this is a non-trivial integral, we may attempt to solve it numerically or with approximation methods. Let's go ahead and compute this numerically.

I'll calculate the value of the integral numerically.The value of the integral

\[
\int_1^6 \sqrt{\frac{9}{x^2} + \frac{x^2}{133} + \frac{1}{2}} \, dx
\]

is approximately \( 6.85 \), with an error on the order of \( 10^{-11} \), which indicates a high degree of accuracy in the numerical result.

Would you like further details on any steps or methods? Here are some related questions:

1. How can this integral be solved using substitution or approximation?
2. What other methods could simplify integrals with radicals?
3. How does the presence of constants like \(\frac{1}{2}\) affect the integrand?
4. How does numerical integration compare to symbolic integration?
5. Can integrals involving \(x^{-2}\) have physical interpretations?

**Tip:** When dealing with complicated integrals, consider applying approximation methods like numerical integration if no simple antiderivative exists.

Solve the integral of sqrt(9/x^2 + x^2/133 + 1/2) from 1 to 6 using numerical methods. This problem explores advanced calculus topics such as radical functions and numerical integration techniques. Suitable for university-level calculus students.

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