`int(2x^(2)+3)/((x^(2)-1)(x^(2)+4))dx=alog((x+1)/(x-1))+b"tan"^(-1)(x)/(2)` , then (a,b) is

Options

A. `(-1//2,1//2)`
B. `(1//2,1//2)`
C. `(-1,1)`
D. `(1,-1)`

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. We can express the integrand as:

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

### Step 1: Set up the equation
We start by multiplying both sides by the denominator \((x^2 - 1)(x^2 + 4)\):

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

### Step 2: Expand the right-hand side
Expanding the right-hand side gives:

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

This simplifies to:

\[ 2x^2 + 3 = Ax^2 + 4A + Bx^3 - Bx + Cx^2 - C. \]

Combining like terms, we have:

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

### Step 3: Equate coefficients
Now, we equate the coefficients from both sides:

1. Coefficient of \(x^3\): \(B = 0\)
2. Coefficient of \(x^2\): \(A + C = 2\)
3. Coefficient of \(x\): \(-B = 0\) (already satisfied)
4. Constant term: \(4A - C = 3\)

### Step 4: Solve the system of equations
From \(B = 0\), we substitute into the other equations:

1. From \(A + C = 2\), we have \(C = 2 - A\).
2. Substitute \(C\) into \(4A - C = 3\):

\[ 4A - (2 - A) = 3 \implies 4A - 2 + A = 3 \implies 5A = 5 \implies A = 1. \]

Now substitute \(A = 1\) back to find \(C\):

\[ C = 2 - A = 2 - 1 = 1. \]

### Step 5: Values of A and C
Now we have \(A = 1\), \(B = 0\), and \(C = 1\).

### Step 6: Rewrite the integral
Now we can rewrite the integral:

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

### Step 7: Integrate
The first integral can be solved using the formula for the integral of a rational function:

\[ \int \frac{1}{x^2 - 1} \, dx = \log \left| \frac{x + 1}{x - 1} \right| + C_1, \]

and the second integral:

\[ \int \frac{1}{x^2 + 4} \, dx = \frac{1}{2} \tan^{-1} \left( \frac{x}{2} \right) + C_2. \]

### Step 8: Combine results
Thus, the complete integral is:

\[ \int \frac{2x^2 + 3}{(x^2 - 1)(x^2 + 4)} \, dx = \log \left| \frac{x + 1}{x - 1} \right| + \frac{1}{2} \tan^{-1} \left( \frac{x}{2} \right) + C. \]

### Step 9: Identify constants a and b
From the given expression \(a \log \left( \frac{x + 1}{x - 1} \right) + b \tan^{-1} \left( x \right)/2\), we can identify:

- \(a = 1\)
- \(b = 1\)

### Final Answer
Thus, the values of \(a\) and \(b\) are:

\[ (a, b) = (1, 1). \]