If `int (dx)/((x^(2)+a^(2))^(3))=(x)/(4a^(2)(x^(2)+a^(2)))+(m)/(na^(2)){(x)/(2a^(2)(x^(2)+a^(2)))+(1)/(2a^(3))tan^(-1)(x/a)}+C,` then `|m-n|` is equal to

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the given integral and find the values of \( m \) and \( n \), we start with the integral:

\[ \int \frac{dx}{(x^2 + a^2)^3} \]

### Step 1: Use the substitution
Let \( x = a \tan \theta \). Then, we have:

\[ dx = a \sec^2 \theta \, d\theta \]

Substituting \( x \) and \( dx \) into the integral, we get:

\[ \int \frac{a \sec^2 \theta \, d\theta}{(a^2 \tan^2 \theta + a^2)^3} \]

### Step 2: Simplify the integral
The expression \( a^2 \tan^2 \theta + a^2 \) can be factored out:

\[ = a^2 (\tan^2 \theta + 1) = a^2 \sec^2 \theta \]

Thus, the integral becomes:

\[ \int \frac{a \sec^2 \theta \, d\theta}{(a^2 \sec^2 \theta)^3} = \int \frac{a \sec^2 \theta \, d\theta}{a^6 \sec^6 \theta} = \frac{1}{a^5} \int \cos^4 \theta \, d\theta \]

### Step 3: Use the identity for \( \cos^4 \theta \)
Using the identity \( \cos^4 \theta = \left(\frac{1 + \cos 2\theta}{2}\right)^2 \):

\[ \cos^4 \theta = \frac{1 + 2\cos 2\theta + \cos^2 2\theta}{4} = \frac{1 + 2\cos 2\theta + \frac{1 + \cos 4\theta}{2}}{4} \]

This simplifies to:

\[ \cos^4 \theta = \frac{3 + 4\cos 2\theta + \cos 4\theta}{8} \]

### Step 4: Integrate \( \cos^4 \theta \)
Now we can integrate:

\[ \int \cos^4 \theta \, d\theta = \frac{1}{8} \int (3 + 4\cos 2\theta + \cos 4\theta) \, d\theta = \frac{3\theta}{8} + 2\sin 2\theta + \frac{\sin 4\theta}{32} + C \]

### Step 5: Substitute back to \( x \)
Now we substitute back \( \theta = \tan^{-1} \left(\frac{x}{a}\right) \):

\[ = \frac{3}{8} \tan^{-1} \left(\frac{x}{a}\right) + 2\sin \left(2 \tan^{-1} \left(\frac{x}{a}\right)\right) + \frac{\sin \left(4 \tan^{-1} \left(\frac{x}{a}\right)\right)}{32} + C \]

### Step 6: Compare with the given expression
The expression we have is:

\[ \frac{x}{4a^2(x^2 + a^2)} + \frac{m}{na^2} \left(\frac{x}{2a^2(x^2 + a^2)}\right) + \frac{1}{2a^3} \tan^{-1} \left(\frac{x}{a}\right) + C \]

From the integration, we can compare the coefficients of \( \tan^{-1} \left(\frac{x}{a}\right) \) and find:

- The coefficient of \( \tan^{-1} \left(\frac{x}{a}\right) \) gives us \( \frac{3}{8} = \frac{1}{2a^3} \) leading to \( m = 3 \) and \( n = 4 \).

### Step 7: Calculate \( |m - n| \)
Thus, we find:

\[ |m - n| = |3 - 4| = 1 \]

### Final Answer
The value of \( |m - n| \) is \( \boxed{1} \).