2 Let $$ f $$ be the function given by $$ f(x)=3 \ln \left(2+x^{2}\right) \cos x $$. What is the average value of fon the closed interval $$ 2 \leq x \leq 6 $$ ? (A) -6.691 (B) -1.673 (C) 3.179 (D) 4.121 (Deview (D) (D) (A)

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It seems that I am unable to directly compute the integral of the function $$ f(x) = 3 \ln(2 + x^2) \cos x $$ using the tools available. However, I can guide you through the process of calculating it manually or using numerical methods.

### Steps to Calculate the Average Value

1. **Set up the integral**:
   $$
   \int_2^6 3 \ln(2 + x^2) \cos x \, dx
   $$

2. **Use numerical integration**:
   Since the integral may not have a simple antiderivative, you can use numerical methods such as Simpson's rule or the trapezoidal rule to approximate the integral.

3. **Calculate the average value**:
   Once you have the value of the integral, you can find the average value using:
   $$
   \text{Average value} = \frac{1}{4} \int_2^6 3 \ln(2 + x^2) \cos x \, dx
   $$

### Numerical Integration Example
If you have access to a calculator or software that can perform numerical integration, you can compute the integral directly.

For example, using a numerical integration tool, you would input:
$$
\int_2^6 3 \ln(2 + x^2) \cos x \, dx
$$

### Final Calculation
After obtaining the integral value, multiply it by $$ \frac{1}{4} $$ to find the average value.

If you would like, I can help you with a specific numerical method or guide you through using a calculator or software to find the integral. Please let me know how you would like to proceed!
The average value of the function $$ f(x) = 3 \ln(2 + x^2) \cos x $$ on the interval $$ [2, 6] $$ is approximately -1.673. **Answer: (B) -1.673**

2
Let be the function given by . What is the average value of fon the closed
interval ?
(A) -6.691
(B) -1.673
© 3.179
(D) 4.121
(Deview
(D)
(D)
(A)

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2 Let be the function given by . What is the average value of fon the closed interval ? (A) -6.691 (B) -1.673 © 3.179 (D) 4.121 (Deview (D) (D) (A)