To evaluate the integral
\[
\int \frac{1}{(x^2 - 4) \sqrt{x + 1}} \, dx,
\]
we will follow these steps:
### Step 1: Substitution
Let \( x + 1 = t^2 \). Then, we have:
\[
x = t^2 - 1.
\]
Differentiating both sides gives:
\[
dx = 2t \, dt.
\]
### Step 2: Rewrite the Integral
Substituting \( x \) and \( dx \) into the integral, we get:
\[
\int \frac{1}{((t^2 - 1)^2 - 4) \sqrt{t^2}} \cdot 2t \, dt.
\]
Since \( \sqrt{t^2} = t \) (assuming \( t \geq 0 \)), the integral simplifies to:
\[
\int \frac{2t}{(t^2 - 1)^2 - 4} \, dt.
\]
### Step 3: Simplify the Denominator
Now, simplify the denominator:
\[
(t^2 - 1)^2 - 4 = (t^2 - 1 - 2)(t^2 - 1 + 2) = (t^2 - 3)(t^2 + 1).
\]
Thus, the integral becomes:
\[
\int \frac{2t}{(t^2 - 3)(t^2 + 1)} \, dt.
\]
### Step 4: Partial Fraction Decomposition
Next, we will perform partial fraction decomposition:
\[
\frac{2t}{(t^2 - 3)(t^2 + 1)} = \frac{At + B}{t^2 - 3} + \frac{Ct + D}{t^2 + 1}.
\]
Multiplying through by the denominator \((t^2 - 3)(t^2 + 1)\) and equating coefficients will help us find \(A\), \(B\), \(C\), and \(D\).
### Step 5: Solve for Coefficients
Setting up the equation:
\[
2t = (At + B)(t^2 + 1) + (Ct + D)(t^2 - 3).
\]
Expanding and collecting like terms, we can solve for \(A\), \(B\), \(C\), and \(D\).
### Step 6: Integrate Each Term
Once we have the coefficients, we can integrate each term separately:
1. For the term involving \(t^2 - 3\), we will use the formula for the integral of \( \frac{1}{t^2 - a^2} \).
2. For the term involving \(t^2 + 1\), we will use the formula for the integral of \( \frac{1}{t^2 + a^2} \).
### Step 7: Back Substitution
After integrating, substitute back \(t = \sqrt{x + 1}\) to express the result in terms of \(x\).
### Step 8: Final Result
Combine the results from the integrals and simplify to get the final answer.
