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

$$$$

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?

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.