`4 int (sqrt(a^(6)+x^(8)))/(x) dx` is equal to

Options

A. `sqrt(a^(6)+x^(8))+(a^(3))/(2)ln|(sqrt(a^(6)+x^(6))-a^(3))/(sqrt(a^(6)+x^(8))-a^(3))|+c`
B. `a^(6)ln|(sqrt(a^(6)+x^(6))-a^(3))/(sqrt(a^(6)+x^(8))+a^(3))|+c`
C. `sqrt(a^(6)+x^(8))+(a^(3))/(2)ln|(sqrt(a^(6)+x^(6))-a^(3))/(sqrt(a^(6)+x^(8))+a^(3))|+c`
D. `a^(6)ln|(sqrt(a^(6)+x^(6))-a^(3))/(sqrt(a^(6)+x^(8))+a^(3))|+c`

Answer

To solve the integral \( 4 \int \frac{\sqrt{a^6 + x^8}}{x} \, dx \), we can follow these steps:

### Step 1: Set up the integral
Let \( I = 4 \int \frac{\sqrt{a^6 + x^8}}{x} \, dx \).

### Step 2: Substitution
We will use the substitution \( t^2 = a^6 + x^8 \). Differentiating both sides gives:
\[ 2t \, dt = 8x^7 \, dx \quad \Rightarrow \quad dx = \frac{t \, dt}{4x^7} \]

### Step 3: Rewrite the integral
Substituting \( dx \) into the integral, we have:
\[ I = 4 \int \frac{\sqrt{t^2}}{x} \cdot \frac{t \, dt}{4x^7} \]
This simplifies to:
\[ I = \int \frac{t^2}{x^8} \, dt \]

### Step 4: Express \( x^8 \) in terms of \( t \)
From our substitution \( t^2 = a^6 + x^8 \), we can express \( x^8 \) as:
\[ x^8 = t^2 - a^6 \]
Thus, the integral becomes:
\[ I = \int \frac{t^2}{t^2 - a^6} \, dt \]

### Step 5: Separate the integral
We can separate the integral:
\[ I = \int \left( 1 + \frac{a^6}{t^2 - a^6} \right) dt \]
This gives us:
\[ I = \int dt + a^6 \int \frac{1}{t^2 - a^6} \, dt \]

### Step 6: Integrate
The first integral is straightforward:
\[ \int dt = t \]
The second integral can be solved using the formula for the integral of \( \frac{1}{x^2 - a^2} \):
\[ \int \frac{1}{x^2 - a^2} \, dx = \frac{1}{2a} \ln \left| \frac{x - a}{x + a} \right| + C \]
So we have:
\[ I = t + a^6 \cdot \frac{1}{2\sqrt{a^6}} \ln \left| \frac{t - \sqrt{a^6}}{t + \sqrt{a^6}} \right| + C \]

### Step 7: Substitute back for \( t \)
Recall that \( t = \sqrt{a^6 + x^8} \), so substituting back gives:
\[ I = \sqrt{a^6 + x^8} + \frac{a^6}{2\sqrt{a^6}} \ln \left| \frac{\sqrt{a^6 + x^8} - \sqrt{a^6}}{\sqrt{a^6 + x^8} + \sqrt{a^6}} \right| + C \]

### Final Answer
Thus, the final answer for the integral \( 4 \int \frac{\sqrt{a^6 + x^8}}{x} \, dx \) is:
\[ I = \sqrt{a^6 + x^8} + \frac{a^6}{2\sqrt{a^6}} \ln \left| \frac{\sqrt{a^6 + x^8} - \sqrt{a^6}}{\sqrt{a^6 + x^8} + \sqrt{a^6}} \right| + C \]