`int_(2)^(3)(x)/((x+2)(x+3))dx=`

Options

A. `log((6^(3).4^(2))/(5^(5)))`
B. `log((3^(6).2^(4))/(5^(5)))`
C. `log((24)/(5))`
D. `log((4^(2).5^(3))/(6^(4)))`

Answer

To solve the definite integral \( I = \int_{2}^{3} \frac{x}{(x+2)(x+3)} \, dx \), we will follow these steps:

### Step 1: Rewrite the integrand
We can rewrite the integrand by adding and subtracting 2 in the numerator:
\[ I = \int_{2}^{3} \frac{x + 2 - 2}{(x+2)(x+3)} \, dx = \int_{2}^{3} \frac{x + 2}{(x+2)(x+3)} \, dx - \int_{2}^{3} \frac{2}{(x+2)(x+3)} \, dx \]

### Step 2: Simplify the first term
The first term simplifies because \( \frac{x + 2}{(x + 2)(x + 3)} = \frac{1}{x + 3} \):
\[ I = \int_{2}^{3} \frac{1}{x + 3} \, dx - \int_{2}^{3} \frac{2}{(x + 2)(x + 3)} \, dx \]

### Step 3: Evaluate the first integral
The first integral can be evaluated:
\[ \int \frac{1}{x + 3} \, dx = \ln |x + 3| + C \]
Thus,
\[ \int_{2}^{3} \frac{1}{x + 3} \, dx = \left[ \ln |x + 3| \right]_{2}^{3} = \ln(6) - \ln(5) = \ln\left(\frac{6}{5}\right) \]

### Step 4: Evaluate the second integral
For the second integral, we can use partial fractions:
\[ \frac{2}{(x + 2)(x + 3)} = \frac{A}{x + 2} + \frac{B}{x + 3} \]
Multiplying through by the denominator:
\[ 2 = A(x + 3) + B(x + 2) \]
Setting \( x = -2 \):
\[ 2 = A(1) \Rightarrow A = 2 \]
Setting \( x = -3 \):
\[ 2 = B(-1) \Rightarrow B = -2 \]
Thus,
\[ \frac{2}{(x + 2)(x + 3)} = \frac{2}{x + 2} - \frac{2}{x + 3} \]
Now we can integrate:
\[ \int_{2}^{3} \left( \frac{2}{x + 2} - \frac{2}{x + 3} \right) \, dx = 2 \left( \int_{2}^{3} \frac{1}{x + 2} \, dx - \int_{2}^{3} \frac{1}{x + 3} \, dx \right) \]
Calculating these integrals:
\[ \int_{2}^{3} \frac{1}{x + 2} \, dx = \left[ \ln |x + 2| \right]_{2}^{3} = \ln(5) - \ln(4) = \ln\left(\frac{5}{4}\right) \]
Thus,
\[ \int_{2}^{3} \left( \frac{2}{x + 2} - \frac{2}{x + 3} \right) \, dx = 2 \left( \ln\left(\frac{5}{4}\right) - \ln\left(\frac{6}{5}\right) \right) \]

### Step 5: Combine results
Now we can combine the results:
\[ I = \ln\left(\frac{6}{5}\right) - 2 \left( \ln\left(\frac{5}{4}\right) - \ln\left(\frac{6}{5}\right) \right) \]
This simplifies to:
\[ I = \ln\left(\frac{6}{5}\right) - 2\ln\left(\frac{5}{4}\right) + 2\ln\left(\frac{6}{5}\right) = 3\ln\left(\frac{6}{5}\right) - 2\ln\left(\frac{5}{4}\right) \]

### Step 6: Final simplification
Using the properties of logarithms:
\[ I = \ln\left(\left(\frac{6}{5}\right)^3\right) - \ln\left(\left(\frac{5}{4}\right)^2\right) = \ln\left(\frac{(6^3)}{(5^2)(4^2)}\right) \]
Thus, the final answer is:
\[ I = \ln\left(\frac{216}{100}\right) = \ln\left(\frac{54}{25}\right) \]