For integral `int f(x- a/x)*(1+a/x^(2))dx`, put `x-a/x=t`
For integral `int f(x+a/x)*(1-(a)/(x^(2)))dx`, put `x+a/x =t`
For integral `int f(x^(2)-(a)/(x^(2)))*(x+(a)/(x^(3)))dx`, put `x^(2)-(a)/(x^(2))=t`
For integral `int f(x^(2)+(a)/(x^(2)))*(x-(a)/(x^(3)))dx`, put `x^(2)+(a)/(x^(2))=t`
many integrands can be brought into above forms by suitable reductions or transformations .
`int(x^(4)-2)/(x^(2)sqrt(x^(4)+x^(2)+2))dx`

Options

A. `sqrt(x^(2)+1+(1)/(x^(2)))+C`
B. `sqrt(x^(2)+1+(2)/(x^(2)))+C`
C. `sqrt(x^(2)+(1)/(x^(2)))+C`
D. `sqrt(x^(2)+(2)/(x^(2)))+C`

Answer

To solve the integral

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

we will follow the steps outlined in the video transcript.

### Step 1: Simplify the Integral

First, we will take \(x^2\) common from the denominator:

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

This simplifies to:

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

### Step 2: Rewrite the Denominator

Next, rewrite the denominator:

\[ \sqrt{x^4 + x^2 + 2} = \sqrt{x^4 + x^2 + 2} = \sqrt{(x^2)^2 + (1)^2 + 2} = \sqrt{x^4 + x^2 + 2}. \]

### Step 3: Identify the Form

Now, we can see that the integral resembles the form:

\[ \int f(x^2 + \frac{a}{x^2}) \cdot (x - \frac{a}{x^3}) \, dx. \]

Here, we can set:

\[ t = x^2 + \frac{2}{x^2}. \]

### Step 4: Differentiate to Find \(dt\)

Now, we differentiate \(t\):

\[ dt = \left(2x - \frac{4}{x^3}\right) dx = \left(2x - \frac{4}{x^3}\right) dx. \]

### Step 5: Substitute in the Integral

From the expression for \(dt\), we can express \(dx\):

\[ dx = \frac{dt}{2x - \frac{4}{x^3}}. \]

Substituting back into the integral:

\[ \int \frac{x^2 - \frac{2}{x^2}}{\sqrt{t}} \cdot \frac{dt}{2x - \frac{4}{x^3}}. \]

### Step 6: Integrate

Now, we can integrate:

\[ \int \frac{1}{\sqrt{t}} \, dt = 2\sqrt{t} + C. \]

### Step 7: Substitute Back

Finally, we substitute back for \(t\):

\[ 2\sqrt{x^2 + \frac{2}{x^2}} + C. \]

Thus, the final answer is:

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