`int_(2)^(3)(1)/(x^(2)+5x+6)dx =`

Options

A. `tan^(-1)((25)/(24))`
B. `log((25)/(24))`
C. `sin^(-1)((25)/(24))`
D. `sec^(-1)((25)/(24))`

Answer

To solve the definite integral

\[ \int_{2}^{3} \frac{1}{x^2 + 5x + 6} \, dx, \]

we will follow these steps:

### Step 1: Factor the quadratic expression

The expression in the denominator can be factored. We have:

\[ x^2 + 5x + 6 = (x + 2)(x + 3). \]

### Step 2: Rewrite the integral

Now we can rewrite the integral as:

\[ \int_{2}^{3} \frac{1}{(x + 2)(x + 3)} \, dx. \]

### Step 3: Use partial fraction decomposition

We can express the integrand using partial fractions:

\[ \frac{1}{(x + 2)(x + 3)} = \frac{A}{x + 2} + \frac{B}{x + 3}. \]

Multiplying through by the denominator \((x + 2)(x + 3)\) gives:

\[ 1 = A(x + 3) + B(x + 2). \]

### Step 4: Solve for A and B

Expanding the right side:

\[ 1 = Ax + 3A + Bx + 2B = (A + B)x + (3A + 2B). \]

Setting the coefficients equal, we have:

1. \(A + B = 0\)
2. \(3A + 2B = 1\)

From the first equation, \(B = -A\). Substituting into the second equation:

\[ 3A + 2(-A) = 1 \implies 3A - 2A = 1 \implies A = 1. \]

Thus, \(B = -1\).

### Step 5: Rewrite the integral

Now we can rewrite the integral:

\[ \int_{2}^{3} \left( \frac{1}{x + 2} - \frac{1}{x + 3} \right) \, dx. \]

### Step 6: Integrate term by term

Now we can integrate:

\[ \int_{2}^{3} \frac{1}{x + 2} \, dx - \int_{2}^{3} \frac{1}{x + 3} \, dx. \]

The integrals are:

\[ \int \frac{1}{x + 2} \, dx = \ln |x + 2| + C, \]
\[ \int \frac{1}{x + 3} \, dx = \ln |x + 3| + C. \]

### Step 7: Evaluate the definite integrals

Evaluating the first integral from 2 to 3:

\[ \left[ \ln |x + 2| \right]_{2}^{3} = \ln(5) - \ln(4). \]

Evaluating the second integral from 2 to 3:

\[ \left[ \ln |x + 3| \right]_{2}^{3} = \ln(6) - \ln(5). \]

### Step 8: Combine results

Putting it all together:

\[ \ln(5) - \ln(4) - (\ln(6) - \ln(5)) = \ln(5) - \ln(4) - \ln(6) + \ln(5). \]

This simplifies to:

\[ 2\ln(5) - \ln(4) - \ln(6). \]

Using the property of logarithms \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\):

\[ = \ln\left(\frac{25}{24}\right). \]

### Final Answer

Thus, the value of the definite integral is:

\[ \int_{2}^{3} \frac{1}{x^2 + 5x + 6} \, dx = \ln\left(\frac{25}{24}\right). \]

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