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solve thes3 2 questions, provide detailed steps and explanations and final answer

Let's solve each question one by one with detailed explanations.

### Question 1
The problem states:
> Which of the following integration formulas can be derived from the differentiation formula
> \[
> \frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1 - x^2}}?
> \]

The differentiation formula given implies that if we integrate the right side, we should get the inverse sine function back:
\[
\int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1}(x) + C
\]
where \(C\) is the constant of integration.

Let's analyze the options provided:
1. **Option 1:** \(\int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1}(x) + C\)
2. **Option 2:** \(\int \sin^{-1}(x) \, dx = \frac{1}{\sqrt{1 - x^2}} + C\)
3. **Option 3:** \(\int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1}(x) + C\)
4. **Option 4:** \(\int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1}(x)\)

Clearly, the correct answer is **Option 1** (and also matches Option 3), as it is the only option that states:
\[
\int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1}(x) + C
\]
So, **Option 1** (and equivalently **Option 3**) is the answer.

### Question 2
This question asks us to use the integration formula:
\[
\int \frac{f'(x)}{f(x)} \, dx = \ln |f(x)| + C
\]
to evaluate the integral:
\[
\int \frac{-4x^3}{x^4 - 1} \, dx
\]

#### Step-by-Step Solution
1. **Identify \( f(x) \) and \( f'(x) \):**
   We can see that in the integrand, \( f(x) = x^4 - 1 \). The derivative of \( f(x) \) is:
   \[
   f'(x) = 4x^3
   \]
   Since the integrand has \(-4x^3\) in the numerator, we can rewrite the integral as:
   \[
   \int \frac{-4x^3}{x^4 - 1} \, dx = \int \frac{f'(x)}{f(x)} \, dx
   \]
   where \( f(x) = x^4 - 1 \) and \( f'(x) = -4x^3 \).

2. **Apply the formula:**
   Using the formula \(\int \frac{f'(x)}{f(x)} \, dx = \ln |f(x)| + C\), we find:
   \[
   \int \frac{-4x^3}{x^4 - 1} \, dx = \ln |x^4 - 1| + C
   \]

3. **Determine the correct answer from the options:**
   The correct answer is therefore:
   \[
   \ln |x^4 - 1| + C
   \]

So, the answer to Question 2 is the option:
\[
\ln |x^4 - 1| + C
\]

### Final Answers
1. **Question 1 Answer:** Option 1 (or Option 3): \(\int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1}(x) + C\)
2. **Question 2 Answer:** \(\ln |x^4 - 1| + C\)

Would you like further details or have any questions about these solutions?

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#### Additional Related Questions
1. What is the general formula for integrating \(\int \frac{f'(x)}{f(x)} \, dx\)?
2. How would the solution change in Question 2 if the numerator was \(4x^3\) instead of \(-4x^3\)?
3. What is the relationship between inverse trigonometric functions and their derivatives?
4. Can we apply the integration formula used in Question 1 to functions other than \(\sin^{-1}(x)\)?
5. Why is the constant of integration important in indefinite integrals?

#### Tip
When working with integrals that involve inverse trigonometric functions, always look for patterns in the derivatives of these functions, as they often simplify the problem.

solve these 2 questions, provide detailed steps and explanations and final answer

Detailed solutions for integrating inverse trigonometric and logarithmic functions. Includes step-by-step explanations on using the differentiation of inverse sine and the logarithmic integration formula. Suitable for undergraduate calculus students.

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