If : `int(2x^(2)+3)/((x^(2)-1)(x^(2)-4))dx=log[((x-2)/(x+))^(a).((x+1)/(x-1))^(b)]+c` then : `(a, b)-=`

Answer

To solve the integral

\[ \int \frac{2x^2 + 3}{(x^2 - 1)(x^2 - 4)} \, dx, \]

we will use partial fraction decomposition.

### Step 1: Set up the partial fraction decomposition

We can express the integrand as:

\[ \frac{2x^2 + 3}{(x^2 - 1)(x^2 - 4)} = \frac{A}{x^2 - 1} + \frac{B}{x^2 - 4}. \]

### Step 2: Multiply through by the denominator

Multiplying both sides by \((x^2 - 1)(x^2 - 4)\) gives:

\[ 2x^2 + 3 = A(x^2 - 4) + B(x^2 - 1). \]

### Step 3: Expand the right-hand side

Expanding the right-hand side:

\[ 2x^2 + 3 = Ax^2 - 4A + Bx^2 - B = (A + B)x^2 + (-4A - B). \]

### Step 4: Set up equations for coefficients

Now, we can equate the coefficients:

1. For \(x^2\): \(A + B = 2\)
2. For the constant term: \(-4A - B = 3\)

### Step 5: Solve the system of equations

From the first equation, we can express \(B\) in terms of \(A\):

\[ B = 2 - A. \]

Substituting this into the second equation:

\[ -4A - (2 - A) = 3 \implies -4A - 2 + A = 3 \implies -3A = 5 \implies A = -\frac{5}{3}. \]

Now substituting \(A\) back to find \(B\):

\[ B = 2 - \left(-\frac{5}{3}\right) = 2 + \frac{5}{3} = \frac{6}{3} + \frac{5}{3} = \frac{11}{3}. \]

### Step 6: Rewrite the integral

Now we can rewrite the integral:

\[ \int \left( \frac{-\frac{5}{3}}{x^2 - 1} + \frac{\frac{11}{3}}{x^2 - 4} \right) dx. \]

### Step 7: Integrate each term

Now we can integrate each term separately:

1. For \(\int \frac{-\frac{5}{3}}{x^2 - 1} \, dx\):
\[ = -\frac{5}{3} \cdot \frac{1}{2} \log \left| \frac{x - 1}{x + 1} \right| = -\frac{5}{6} \log \left| \frac{x - 1}{x + 1} \right|. \]

2. For \(\int \frac{\frac{11}{3}}{x^2 - 4} \, dx\):
\[ = \frac{11}{3} \cdot \frac{1}{2} \log \left| \frac{x - 2}{x + 2} \right| = \frac{11}{6} \log \left| \frac{x - 2}{x + 2} \right|. \]

### Step 8: Combine results

Combining these results, we have:

\[ -\frac{5}{6} \log \left| \frac{x - 1}{x + 1} \right| + \frac{11}{6} \log \left| \frac{x - 2}{x + 2} \right| + C. \]

### Step 9: Rewrite in the required form

Using properties of logarithms, we can combine these logarithms:

\[ = \log \left( \left( \frac{x - 2}{x + 2} \right)^{\frac{11}{6}} \cdot \left( \frac{x + 1}{x - 1} \right)^{-\frac{5}{6}} \right) + C. \]

### Step 10: Identify values of \(a\) and \(b\)

From the expression, we can identify:

- \(a = \frac{11}{6}\)
- \(b = -\frac{5}{6}\)

Thus, the final answer is:

\[ (a, b) = \left(\frac{11}{6}, -\frac{5}{6}\right). \]