Question

The integral of \( \int_{-2}^{2} x^4 \, dx \) denominator \( (1+5x^2) \) is:

• A. 0
• B. 4/3
• C. 32/5
• D. 64/5

Hint:

For symmetric intervals \([-a, a]\), the integral of an even function (like \( x^4 \)) is \( 2 \int_{0}^{a} f(x) \, dx \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
For rational functions where the degree of the numerator is greater than the degree of the denominator, perform polynomial division.

Step 2: Detailed Explanation:
\( \frac{x^4}{5x^2 + 1} = \frac{1}{5} x^2 - \frac{1}{25} + \frac{1/25}{5x^2 + 1} \).
Integrating \( \frac{1}{5} x^2 - \frac{1}{25} + \frac{1}{25(5x^2 + 1)} \) from 0 to 2:
\( [ \frac{x^3}{15} - \frac{x}{25} + \frac{1}{25\sqrt{5}} \arctan(\sqrt{5}x) ]_0^2 \).
Evaluating at bounds gives the specific numerical result (approximately 4/3).

Step 3: Final Answer:
The integral evaluates to 4/3.