The mean value of the function `f(x) = (1)/( x^2 + x)` on the interval `[1, 3//2]` is

Options

A. `log (6//5)`
B. `2 log (6//5)`
C. `4`
D. `log 3//5`

Answer

To find the mean value of the function \( f(x) = \frac{1}{x^2 + x} \) on the interval \([1, \frac{3}{2}]\), we will use the formula for the mean value of a function over an interval \([a, b]\):

\[ \text{Mean Value} = \frac{1}{b - a} \int_a^b f(x) \, dx \]

### Step 1: Identify the interval and function
Here, \( a = 1 \) and \( b = \frac{3}{2} \). The function is \( f(x) = \frac{1}{x^2 + x} \).

### Step 2: Calculate \( b - a \)
\[ b - a = \frac{3}{2} - 1 = \frac{1}{2} \]

### Step 3: Set up the integral
We need to compute the integral:
\[ \int_1^{\frac{3}{2}} \frac{1}{x^2 + x} \, dx \]

### Step 4: Simplify the integrand
We can factor the denominator:
\[ x^2 + x = x(x + 1) \]
Thus, we can rewrite the integrand:
\[ \frac{1}{x^2 + x} = \frac{1}{x(x + 1)} \]

### Step 5: Use partial fraction decomposition
We can express \( \frac{1}{x(x + 1)} \) as:
\[ \frac{1}{x(x + 1)} = \frac{A}{x} + \frac{B}{x + 1} \]
Multiplying through by the denominator \( x(x + 1) \):
\[ 1 = A(x + 1) + Bx \]
Setting \( x = 0 \) gives \( A = 1 \). Setting \( x = -1 \) gives \( B = -1 \). Thus:
\[ \frac{1}{x(x + 1)} = \frac{1}{x} - \frac{1}{x + 1} \]

### Step 6: Set up the integral with partial fractions
Now we can rewrite the integral:
\[ \int_1^{\frac{3}{2}} \left( \frac{1}{x} - \frac{1}{x + 1} \right) \, dx \]

### Step 7: Integrate
Now we integrate each term:
\[ \int \frac{1}{x} \, dx = \ln |x| \quad \text{and} \quad \int \frac{1}{x + 1} \, dx = \ln |x + 1| \]
Thus, we have:
\[ \int_1^{\frac{3}{2}} \left( \frac{1}{x} - \frac{1}{x + 1} \right) \, dx = \left[ \ln |x| - \ln |x + 1| \right]_1^{\frac{3}{2}} \]

### Step 8: Evaluate the definite integral
Calculating the limits:
\[ \left[ \ln \left( \frac{3}{2} \right) - \ln \left( \frac{5}{2} \right) \right] - \left[ \ln(1) - \ln(2) \right] \]
Since \( \ln(1) = 0 \):
\[ = \ln \left( \frac{3/2}{5/2} \right) + \ln(2) = \ln \left( \frac{3}{5} \cdot 2 \right) = \ln \left( \frac{6}{5} \right) \]

### Step 9: Substitute back into the mean value formula
Now substituting back into the mean value formula:
\[ \text{Mean Value} = \frac{1}{\frac{1}{2}} \cdot \ln \left( \frac{6}{5} \right) = 2 \ln \left( \frac{6}{5} \right) \]

### Final Answer
Thus, the mean value of the function \( f(x) \) on the interval \([1, \frac{3}{2}]\) is:
\[ 2 \ln \left( \frac{6}{5} \right) \]