If `intsqrt((x^(4))/(a^(6)+x^(6)))dx=g (x)+C`,then g (x)=

Options

A. `(1)/(3)log|x^(3)-sqrt(a^(6)+x^(6))|`
B. `log|x^(3)+sqrt(a^(6)+x^(6))|`
C. `(1)/(3)log|x^(3)+sqrt(a^(6)+x^(6))|`
D. none of these

Explanation

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

### Step 1: Rewrite the Integral
We start by rewriting the integral:
\[ \int \sqrt{\frac{x^4}{a^6 + x^6}} \, dx = \int \frac{x^2}{\sqrt{a^6 + x^6}} \, dx \]

### Step 2: Substitution
Next, we will use the substitution \( x^3 = t \). Therefore, differentiating both sides gives us:
\[ 3x^2 \, dx = dt \quad \Rightarrow \quad x^2 \, dx = \frac{dt}{3} \]
Now, substituting \( x^3 = t \) into the integral:
\[ \int \frac{x^2}{\sqrt{a^6 + x^6}} \, dx = \int \frac{\frac{dt}{3}}{\sqrt{a^6 + t^2}} = \frac{1}{3} \int \frac{dt}{\sqrt{a^6 + t^2}} \]

### Step 3: Integral of the Form
The integral \( \int \frac{dt}{\sqrt{a^6 + t^2}} \) is of the standard form \( \int \frac{dx}{\sqrt{a^2 + x^2}} = \log |x + \sqrt{a^2 + x^2}| + C \). Here, we can identify \( a^2 = a^6 \) and \( x = t \):
\[ \int \frac{dt}{\sqrt{a^6 + t^2}} = \log |t + \sqrt{a^6 + t^2}| + C \]

### Step 4: Substitute Back
Now, substituting back \( t = x^3 \):
\[ \frac{1}{3} \left( \log |x^3 + \sqrt{a^6 + (x^3)^2}| + C \right) = \frac{1}{3} \log |x^3 + \sqrt{a^6 + x^6}| + C \]

### Final Result
Thus, we have:
\[ g(x) = \frac{1}{3} \log |x^3 + \sqrt{a^6 + x^6}| \]