`int (x + (1)/(x)) ^(n +5) ((x ^(2) -1)/( x ^(2))) dx` is equal to:

Options

A. `((x +(1)/(x) ) ^(n + 6))/( n + 6) +c`
B. `[ (x ^(2) + 1)/( x ^(2)) ] ^(n + 6)( n+ 6) + c`
C. `[ (x)/( x ^(2) + 1) ] ^(n + 6) (n+6) + c`
D. None of these

Answer

To solve the integral

\[ I = \int \left( x + \frac{1}{x} \right)^{n+5} \cdot \frac{x^2 - 1}{x^2} \, dx, \]

we can follow these steps:

### Step 1: Rewrite the Integral
We can rewrite the integral in a more manageable form:

\[ I = \int \left( x + \frac{1}{x} \right)^{n+5} \cdot \left( 1 - \frac{1}{x^2} \right) \, dx. \]

### Step 2: Substitute
Let

\[ y = x + \frac{1}{x}. \]

Then, differentiate \(y\):

\[ \frac{dy}{dx} = 1 - \frac{1}{x^2} \implies dy = \left( 1 - \frac{1}{x^2} \right) dx. \]

This means we can substitute \(dx\) in terms of \(dy\):

\[ dx = \frac{dy}{1 - \frac{1}{x^2}}. \]

### Step 3: Change of Variables
Now we substitute \(y\) and \(dx\) into the integral:

\[ I = \int y^{n+5} \, dy. \]

### Step 4: Integrate
Now we can integrate:

\[ I = \frac{y^{n+6}}{n+6} + C. \]

### Step 5: Substitute Back
Finally, substitute back \(y = x + \frac{1}{x}\):

\[ I = \frac{\left( x + \frac{1}{x} \right)^{n+6}}{n+6} + C. \]

### Final Result
Thus, the integral evaluates to:

\[ I = \frac{\left( x + \frac{1}{x} \right)^{n+6}}{n+6} + C. \]

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