We need to compute the integral:

\[
I = \int_0^1 -9 \sqrt{x^2 + 1} \, dx
\]

We can factor out the constant \(-9\) to simplify the integral:

\[
I = -9 \int_0^1 \sqrt{x^2 + 1} \, dx
\]

This integral can be solved using a trigonometric substitution. We'll use the substitution:

\[
x = \sinh(\theta)
\]

where \(\sinh(\theta)\) is the hyperbolic sine function, since \(\sqrt{x^2 + 1}\) suggests a hyperbolic identity (\(\cosh^2(\theta) = 1 + \sinh^2(\theta)\)).

First, let's compute the substitution step-by-step:

1. \(x = \sinh(\theta)\), so \(dx = \cosh(\theta) \, d\theta\).
2. When \(x = 0\), \(\theta = 0\); and when \(x = 1\), \(\theta = \sinh^{-1}(1)\).
3. \(\sqrt{x^2 + 1} = \cosh(\theta)\), by the identity \(\cosh^2(\theta) = 1 + \sinh^2(\theta)\).

Thus, the integral becomes:

\[
I = -9 \int_0^{\sinh^{-1}(1)} \cosh^2(\theta) \, d\theta
\]

We can use the identity \(\cosh^2(\theta) = \frac{1 + \cosh(2\theta)}{2}\) to simplify the integral:

\[
I = -9 \int_0^{\sinh^{-1}(1)} \frac{1 + \cosh(2\theta)}{2} \, d\theta
\]

Now we can integrate term-by-term:

\[
I = -9 \left[\frac{\theta}{2} + \frac{\sinh(2\theta)}{4} \right]_0^{\sinh^{-1}(1)}
\]

Evaluating this expression at the limits \(\theta = 0\) and \(\theta = \sinh^{-1}(1)\), we get the final result.

Let me calculate this explicitly for you.The value of the integral is approximately:

\[
I \approx -10.33
\]

Would you like further clarification on the steps, or do you have any other questions?

Here are 5 related questions for further exploration:
1. What is the hyperbolic substitution method and when is it useful?
2. How do trigonometric and hyperbolic identities simplify integrals?
3. Can you solve \(\int_0^1 \sqrt{x^2 + 1} \, dx\) without the constant factor?
4. How does substitution affect the limits of integration in definite integrals?
5. How would this result change if the limits of integration were different?

**Tip**: When faced with integrals involving square roots of quadratics, consider using hyperbolic or trigonometric substitutions to simplify.

Learn how to compute the definite integral -9∫_0^1 sqrt(x^2 + 1) dx using hyperbolic substitution. This problem covers integral calculus concepts, including trigonometric substitution, hyperbolic functions, and their applications. Suitable for college-level students or advanced calculus learners.

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