Questions: WeBWorK / MAT 1193 Fall 2024 / Homework 8 Antiderivatives Previous Problem Problem List Next Problem Homework 8 Antiderivatives: (1 point) Antiderivatives Find all possible functions with the given derivative. 1. If y′ = -1/x^2, then y= 2. If y′ = 1 - 1/x^2, then y= 3. If y′ = 6 + 1/x^2, then y= For additional help with this problem type, access the following resources: - Read Chapter 5.2 of the text.

Solution Steps

To find the antiderivative of a function, we need to integrate the given derivative. The antiderivative will include a constant of integration, \( C \), since the derivative of a constant is zero. For each given derivative, we will perform the integration to find the corresponding function.

1. Integrate \( y' = \frac{-1}{x^2} \) to find \( y \).
2. Integrate \( y' = 1 - \frac{1}{x^2} \) to find \( y \).
3. Integrate \( y' = 6 + \frac{1}{x^2} \) to find \( y \).

To find the antiderivative, we integrate: \[ y = \int \frac{-1}{x^2} \, dx = C_1 + \frac{1}{x} \]

Next, we integrate: \[ y = \int \left(1 - \frac{1}{x^2}\right) \, dx = C_2 + x + \frac{1}{x} \]

Finally, we integrate: \[ y = \int \left(6 + \frac{1}{x^2}\right) \, dx = C_3 + 6x - \frac{1}{x} \]

Final Answer