If d/dx (f(x)) = 2x + 3/x and f(1) = 1 then f(x) is

To find \( f(x) \), we need to integrate the given derivative \( \frac{d}{dx} f(x) = 2x + \frac{3}{x} \).

First, we integrate each term separately:
\[ \int (2x + \frac{3}{x}) \, dx \]

1. Integrate \( 2x \):
\[ \int 2x \, dx = x^2 + C_1 \]

2. Integrate \( \frac{3}{x} \):
\[ \int \frac{3}{x} \, dx = 3 \ln |x| + C_2 \]

Combining these results, we get:
\[ f(x) = x^2 + 3 \ln |x| + C \]

Now, we use the initial condition \( f(1) = 1 \) to find the constant \( C \):
\[ f(1) = 1^2 + 3 \ln |1| + C = 1 + 0 + C = 1 \]
\[ C = 0 \]

Therefore, the function \( f(x) \) is:
\[ f(x) = x^2 + 3 \ln |x| \]

Do you want more details or have any questions? Here are 5 related questions you can ask:

1. How do you integrate \( 2x \) step-by-step?
2. Why is \( \ln |x| \) used instead of \( \ln x \)?
3. What are the properties of the natural logarithm function?
4. Can you explain how to use initial conditions to find constants in integrals?
5. What are the rules for integration of basic functions?

**Tip:** Always check initial conditions carefully to ensure the correct determination of constants in indefinite integrals.

Learn how to find the function f(x) from its derivative using integration. The problem involves differentiating and integrating, with an initial condition provided. Suitable for advanced high school or early college students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation