`intsqrt((x^4)/(a^6+x^6))dx`

Answer

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

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

**Hint:** Factor out \( x^4 \) from the square root in the numerator to simplify the expression.

### Step 2: Rewrite the Denominator
Next, we can express the denominator in a more manageable form:
\[ \sqrt{a^6 + x^6} = \sqrt{(x^3)^2 + (a^2)^3} \]
Thus, we rewrite the integral as:
\[ \int \frac{x^2}{\sqrt{(x^3)^2 + (a^2)^3}} \, dx \]

**Hint:** Recognize that this is a form suitable for substitution.

### Step 3: Substitution
Let \( t = x^3 \). Then, we differentiate:
\[ dt = 3x^2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3x^2} \]
Substituting \( x^2 \) in terms of \( t \):
\[ x^2 = (t^{1/3})^2 = t^{2/3} \]
Now, substituting into the integral:
\[ \int \frac{t^{2/3}}{\sqrt{t^2 + a^6}} \cdot \frac{dt}{3t^{2/3}} = \frac{1}{3} \int \frac{dt}{\sqrt{t^2 + a^6}} \]

**Hint:** The substitution simplifies the integral significantly.

### Step 4: Solve the Integral
We now have:
\[ \frac{1}{3} \int \frac{dt}{\sqrt{t^2 + a^6}} \]
This integral is a standard form:
\[ \int \frac{dt}{\sqrt{t^2 + a^2}} = \log |t + \sqrt{t^2 + a^2}| + C \]
Applying this to our integral:
\[ \frac{1}{3} \log |t + \sqrt{t^2 + a^6}| + C \]

**Hint:** Remember to apply the logarithmic identity correctly.

### Step 5: Back Substitute
Now we substitute back \( t = x^3 \):
\[ \frac{1}{3} \log |x^3 + \sqrt{x^6 + a^6}| + C \]

**Hint:** Ensure that you replace \( t \) with its original variable correctly.

### Final Answer
Thus, the final answer is:
\[ \int \sqrt{\frac{x^4}{a^6 + x^6}} \, dx = \frac{1}{3} \log |x^3 + \sqrt{x^6 + a^6}| + C \]