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:
The integral of a function \( f(x) \) from \( a \) to \( b \) is given by \( [F(x)]_a^b = F(b) - F(a) \). For the function \( f(x) = x^4 \), the antiderivative is \( \frac{x^5}{5} \). Note: The provided expression \( \frac{x^4}{1+5x^2} \) does not evaluate to any of the given options. Given the options, it is highly probable that the denominator is a typographical error and the intended integrand is \( x^4 \).

Step 2: Key Formula or Approach:
\( \int_{-2}^{2} x^4 \, dx = \left[ \frac{x^5}{5} \right]_{-2}^{2} \)

Step 3: Detailed Explanation:
\[ \left[ \frac{x^5}{5} \right]_{-2}^{2} = \frac{(2)^5}{5} - \frac{(-2)^5}{5} \]
\[ = \frac{32}{5} - \left( \frac{-32}{5} \right) \]
\[ = \frac{32}{5} + \frac{32}{5} = \frac{64}{5} \]

Step 4: Final Answer:
The value of the integral is \( 64/5 \).