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

Options

A. `((11)/(12), (5)/(6))`
B. `((11)/(12),-(5)/(6))`
C. `(-(11)/(12),(5)/(6))`
D. none of these

Answer

To solve the integral

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

we will use the method of partial fractions. Let's break down the solution step by step.

### Step 1: Rewrite the Integral
We start with the integral:

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

### Step 2: Substitute \( t = x^2 \)
To simplify the expression, we can substitute \( t = x^2 \). This gives us:

\[ \int \frac{2t + 3}{(t - 1)(t - 4)} \cdot \frac{1}{2\sqrt{t}} \, dt \]

However, for the sake of partial fraction decomposition, we will keep it as:

\[ \int \frac{2t + 3}{(t - 1)(t - 4)} \, dt \]

### Step 3: Partial Fraction Decomposition
We express the integrand as:

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

Multiplying through by the denominator \((t - 1)(t - 4)\) gives:

\[ 2t + 3 = A(t - 4) + B(t - 1) \]

### Step 4: Solve for A and B
Expanding the right side:

\[ 2t + 3 = At - 4A + Bt - B \]

Combining like terms:

\[ 2t + 3 = (A + B)t + (-4A - B) \]

Setting coefficients equal gives us the system:

1. \( A + B = 2 \)
2. \( -4A - B = 3 \)

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

\[ B = 2 - A \]

Substituting into the second equation:

\[ -4A - (2 - A) = 3 \]

This simplifies to:

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

Substituting 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 5: Rewrite the Integral
Now we can rewrite the integral:

\[ \int \left( \frac{-5/3}{t - 1} + \frac{11/3}{t - 4} \right) dt \]

### Step 6: Integrate
Integrating term by term:

\[ = -\frac{5}{3} \ln |t - 1| + \frac{11}{3} \ln |t - 4| + C \]

Substituting back \( t = x^2 \):

\[ = -\frac{5}{3} \ln |x^2 - 1| + \frac{11}{3} \ln |x^2 - 4| + C \]

### Step 7: Combine Logarithms
Using properties of logarithms:

\[ = \ln \left( \frac{(x^2 - 4)^{11/3}}{(x^2 - 1)^{5/3}} \right) + C \]

### Step 8: Final Form
Thus, we have:

\[ \int \frac{2x^2 + 3}{(x^2 - 1)(x^2 - 4)} \, dx = \ln \left( \left( \frac{x^2 - 4}{x^2 - 1} \right)^{\frac{11}{3}} \right) + C \]

### Step 9: Identify a and b
From the given equation:

\[ \ln \left( \left( \frac{x - 2}{x + 2} \right)^{a} \cdot \left( \frac{x + 1}{x - 1} \right)^{b} \right) + C \]

We can identify:

- \( a = \frac{11}{3} \)
- \( b = -\frac{5}{3} \)

Thus, the final answer for \( (a, b) \) is:

\[ (a, b) = \left( \frac{11}{3}, -\frac{5}{3} \right) \]