If `int(1)/((x^(2)+4)(x^(2)+9))dx=A" tan"^(-1)(x)/(2)+Btan^(-1)((x)/(3))+C`, then A-B=

Options

A. `(1)/(6)`
B. `(1)/(30)`
C. `-(1)/(30)`
D. `-(1)/(6)`

Explanation

To solve the problem, we need to find the values of constants \( A \) and \( B \) from the given integral and then compute \( A - B \).

### Step 1: Set up the integral
We start with the integral:
\[ \int \frac{1}{(x^2 + 4)(x^2 + 9)} \, dx = A \tan^{-1}\left(\frac{x}{2}\right) + B \tan^{-1}\left(\frac{x}{3}\right) + C \]

### Step 2: Differentiate both sides
Next, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx} \left( \int \frac{1}{(x^2 + 4)(x^2 + 9)} \, dx \right) = \frac{1}{(x^2 + 4)(x^2 + 9)} \]
Using the chain rule on the right side, we have:
\[ \frac{d}{dx} \left( A \tan^{-1}\left(\frac{x}{2}\right) + B \tan^{-1}\left(\frac{x}{3}\right) + C \right) = A \cdot \frac{1}{1 + \left(\frac{x}{2}\right)^2} \cdot \frac{1}{2} + B \cdot \frac{1}{1 + \left(\frac{x}{3}\right)^2} \cdot \frac{1}{3} \]

### Step 3: Simplify the derivatives
This simplifies to:
\[ \frac{A}{2} \cdot \frac{1}{1 + \frac{x^2}{4}} + \frac{B}{3} \cdot \frac{1}{1 + \frac{x^2}{9}} \]
which can be rewritten as:
\[ \frac{A}{2} \cdot \frac{4}{4 + x^2} + \frac{B}{3} \cdot \frac{9}{9 + x^2} \]

### Step 4: Combine the fractions
To combine these fractions, we need a common denominator:
\[ \frac{2A \cdot 4 + 3B \cdot 9}{6(4 + x^2)(9 + x^2)} \]
Setting this equal to the left side:
\[ \frac{1}{(x^2 + 4)(x^2 + 9)} \]

### Step 5: Equate coefficients
Now, we equate the numerators:
\[ 2A \cdot 4 + 3B \cdot 9 = 6 \]

### Step 6: Set up the system of equations
From the previous steps, we also have:
1. \( 2A + 3B = 0 \) (from the coefficients of \( x^2 \))
2. \( 8A + 27B = 6 \) (from the constant term)

### Step 7: Solve the system of equations
From the first equation, we can express \( A \) in terms of \( B \):
\[ A = -\frac{3B}{2} \]
Substituting this into the second equation:
\[ 8\left(-\frac{3B}{2}\right) + 27B = 6 \]
This simplifies to:
\[ -12B + 27B = 6 \implies 15B = 6 \implies B = \frac{2}{5} \]
Now substituting \( B \) back to find \( A \):
\[ A = -\frac{3 \cdot \frac{2}{5}}{2} = -\frac{3}{5} \]

### Step 8: Calculate \( A - B \)
Now we compute \( A - B \):
\[ A - B = -\frac{3}{5} - \frac{2}{5} = -\frac{5}{5} = -1 \]

### Final Answer
Thus, the value of \( A - B \) is:
\[ \boxed{-1} \]