Integrate the following:
a) ∫ 9
√16+𝑥2 𝑑𝑥
b) ∫ 6
√4−(𝑥−1)2 𝑑𝑥

To solve these integrals, we need to apply appropriate integration techniques. Let's start with part (a).

### Part (a): \(\int \frac{9}{\sqrt{16+x^2}} \, dx\)

To integrate this, we can use the standard form of the integral:
\[ \int \frac{1}{\sqrt{a^2+x^2}} \, dx = \sinh^{-1}\left(\frac{x}{a}\right) + C \]

Here, \(a = 4\). Thus, we have:
\[ \int \frac{9}{\sqrt{16+x^2}} \, dx = 9 \int \frac{1}{\sqrt{4^2+x^2}} \, dx = 9 \sinh^{-1}\left(\frac{x}{4}\right) + C \]

So, the integral is:
\[ 9 \sinh^{-1}\left(\frac{x}{4}\right) + C \]

### Part (b): \(\int \frac{6}{\sqrt{4-(x-1)^2}} \, dx\)

To integrate this, we recognize that it fits the form of the standard integral:
\[ \int \frac{1}{\sqrt{a^2 - (x-b)^2}} \, dx = \arcsin\left(\frac{x-b}{a}\right) + C \]

Here, \(a = 2\) and \(b = 1\). Thus, we have:
\[ \int \frac{6}{\sqrt{4-(x-1)^2}} \, dx = 6 \int \frac{1}{\sqrt{2^2-(x-1)^2}} \, dx = 6 \arcsin\left(\frac{x-1}{2}\right) + C \]

So, the integral is:
\[ 6 \arcsin\left(\frac{x-1}{2}\right) + C \]

### Summary

1. \(\int \frac{9}{\sqrt{16+x^2}} \, dx = 9 \sinh^{-1}\left(\frac{x}{4}\right) + C\)
2. \(\int \frac{6}{\sqrt{4-(x-1)^2}} \, dx = 6 \arcsin\left(\frac{x-1}{2}\right) + C\)

Would you like further details or explanations on these integrals? If so, please let me know!

Here are 8 related questions that expand on this information:
1. What is the definition of the inverse hyperbolic sine function, \(\sinh^{-1}(x)\)?
2. How do you derive the formula for the integral of \(\frac{1}{\sqrt{a^2 + x^2}}\)?
3. What are the properties and applications of the inverse trigonometric functions, particularly \(\arcsin(x)\)?
4. How can we use trigonometric substitution to solve integrals involving \(\sqrt{a^2 + x^2}\)?
5. Can we apply similar techniques to integrate functions involving \(\sqrt{x^2 - a^2}\)?
6. How does one handle integrals of the form \(\int \frac{1}{\sqrt{a^2 - x^2}} \, dx\) when \(a < x\)?
7. What are some real-world applications of these integrals in physics or engineering?
8. How do you find the constants of integration when given initial conditions or boundary values?

### Tip:
When facing integrals involving square roots of quadratic expressions, consider using trigonometric or hyperbolic substitutions to simplify the integral.

Learn how to integrate the expressions 9/sqrt(16+x^2) and 6/sqrt(4-(x-1)^2) using trigonometric substitution and hyperbolic functions. This problem covers advanced integration techniques suitable for undergraduate students.

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