Math Problem Statement

what is the f(x) if the first derivative of f(x) is x / (9 - x^4)

Solution

Given that the first derivative of f(x)f(x) is:

f(x)=x9x4f'(x) = \frac{x}{9 - x^4}

To find f(x)f(x), we need to integrate f(x)f'(x) with respect to xx.

Step 1: Set up the integral

f(x)=x9x4dxf(x) = \int \frac{x}{9 - x^4} \, dx

Step 2: Simplify the integral

We notice that the denominator involves x4x^4, and the numerator is xx. A substitution may help simplify the integral.

Let: u=9x4u = 9 - x^4 Then, the derivative of uu with respect to xx is: du=4x3dxdu = -4x^3 \, dx To match the xx term in the numerator, we need to isolate a single xx, so rewrite uu as: [ f(x) = \int \frac{1}{9 - x^4} dx

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Integration
Substitution Method

Formulas

f'(x) = x / (9 - x^4)
Integration formula: ∫ f'(x) dx = f(x)

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12 (Advanced Calculus)