Evaluate : `int(2x+7)/(x^(2)-x-2)dx`

Answer

To evaluate the integral \( \int \frac{2x + 7}{x^2 - x - 2} \, dx \), we can follow these steps:

### Step 1: Factor the Denominator
First, we need to factor the quadratic expression in the denominator \( x^2 - x - 2 \).

The factors of \( x^2 - x - 2 \) can be found by looking for two numbers that multiply to \(-2\) and add to \(-1\). These numbers are \(-2\) and \(1\).

Thus, we can factor it as:
\[ x^2 - x - 2 = (x - 2)(x + 1) \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral:
\[ \int \frac{2x + 7}{(x - 2)(x + 1)} \, dx \]

### Step 3: Partial Fraction Decomposition
Next, we will use partial fraction decomposition to express \( \frac{2x + 7}{(x - 2)(x + 1)} \) in a simpler form:
\[ \frac{2x + 7}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1} \]
Multiplying through by the denominator \((x - 2)(x + 1)\) gives:
\[ 2x + 7 = A(x + 1) + B(x - 2) \]

### Step 4: Solve for A and B
Expanding the right-hand side:
\[ 2x + 7 = Ax + A + Bx - 2B = (A + B)x + (A - 2B) \]
Now, we can equate coefficients:
1. \( A + B = 2 \)
2. \( A - 2B = 7 \)

From the first equation, we can express \( A \) in terms of \( B \):
\[ A = 2 - B \]
Substituting into the second equation:
\[ (2 - B) - 2B = 7 \implies 2 - 3B = 7 \implies -3B = 5 \implies B = -\frac{5}{3} \]
Now substituting back to find \( A \):
\[ A = 2 - \left(-\frac{5}{3}\right) = 2 + \frac{5}{3} = \frac{6}{3} + \frac{5}{3} = \frac{11}{3} \]

### Step 5: Rewrite the Integral
Now we can rewrite the integral as:
\[ \int \left( \frac{11/3}{x - 2} - \frac{5/3}{x + 1} \right) \, dx \]

### Step 6: Integrate
Now we can integrate term by term:
\[ \int \frac{11/3}{x - 2} \, dx - \int \frac{5/3}{x + 1} \, dx \]
This results in:
\[ \frac{11}{3} \ln |x - 2| - \frac{5}{3} \ln |x + 1| + C \]

### Step 7: Combine the Logarithms
Using the properties of logarithms, we can combine the logarithmic terms:
\[ = \frac{11}{3} \ln |x - 2| - \frac{5}{3} \ln |x + 1| + C = \frac{1}{3} \ln \left( \frac{(x - 2)^{11}}{(x + 1)^{5}} \right) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{2x + 7}{x^2 - x - 2} \, dx = \frac{1}{3} \ln \left( \frac{(x - 2)^{11}}{(x + 1)^{5}} \right) + C \]