To find the integral of $g(x)$ from 1 to 5, where $g(x)$ is the inverse function of $f(x) = x^3 + 3x + 1$, we can use a known result involving inverse functions and integration:

If $g(x)$ is the inverse of $f(x)$, then: $\int_{a}^{b} g(x) \, dx = \int_{f(a)}^{f(b)} x \, f'(x) \, dx$

Given $f(x) = x^3 + 3x + 1$, let's proceed step-by-step:

Step 1: Calculate $f'(x)$

$f(x) = x^3 + 3x + 1 \implies f'(x) = 3x^2 + 3$

Step 2: Evaluate $f(1)$ and $f(5)$

Calculate $f(1)$ and $f(5)$ to get the limits of integration in terms of $x$ for the integral.

1. $f(1) = 1^3 + 3 \cdot 1 + 1 = 5$
2. $f(5) = 5^3 + 3 \cdot 5 + 1 = 125 + 15 + 1 = 141$

Thus, $\int_{1}^{5} g(x) \, dx = \int_{5}^{141} x \cdot (3x^2 + 3) \, dx$

Step 3: Set up and simplify the integral

$\int_{5}^{141} x \cdot (3x^2 + 3) \, dx = \int_{5}^{141} (3x^3 + 3x) \, dx$

Now we can split this integral and compute each part separately: $= \int_{5}^{141} 3x^3 \, dx + \int_{5}^{141} 3x \, dx$

Would you like me to proceed with calculating each part of this integral?

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Here are 5 additional questions to extend your understanding of this topic:

1. What properties define the inverse of a function, and when do inverses exist?
2. How can derivatives be used to confirm if two functions are inverses?
3. Why is the result used here helpful for integrating inverse functions?
4. What are common challenges when calculating integrals involving inverses?
5. Can this integration approach be applied to trigonometric inverse functions?

Tip: For inverse functions, use the derivative test $f(g(x)) = x$ and $g(f(x)) = x$ to confirm their relationship.