`int (x^5 dx)/(sqrt((1 + x^3))) = `

Options

A. `2/3sqrt((1 + x^3)) (x^2 + 2)`
B. `2/9 sqrt((1 +x^3))(x^3 - 4)`
C. `2/9sqrt((1 + x^3)) (x^2 + 4)`
D. `2/9 sqrt((1 +x^3))(x^3 - 2)`

Answer

To solve the integral \( \int \frac{x^5}{\sqrt{1 + x^3}} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can rewrite \( x^5 \) as \( x^3 \cdot x^2 \). Thus, we have:
\[ \int \frac{x^5}{\sqrt{1 + x^3}} \, dx = \int \frac{x^3 \cdot x^2}{\sqrt{1 + x^3}} \, dx \]

**Hint:** Break down the numerator into simpler parts that can be easily substituted.

### Step 2: Substitution
Let \( t = 1 + x^3 \). Then, differentiate both sides:
\[ dt = 3x^2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3x^2} \]
Now, we also need to express \( x^3 \) in terms of \( t \):
\[ x^3 = t - 1 \]
Substituting these into the integral gives:
\[ \int \frac{(t - 1)x^2}{\sqrt{t}} \cdot \frac{dt}{3x^2} = \frac{1}{3} \int \frac{(t - 1)}{\sqrt{t}} \, dt \]

**Hint:** Use a substitution to simplify the integral.

### Step 3: Simplify the Integral
Now, we can simplify the integral:
\[ \frac{1}{3} \int \left( \frac{t}{\sqrt{t}} - \frac{1}{\sqrt{t}} \right) dt = \frac{1}{3} \int \left( t^{1/2} - t^{-1/2} \right) dt \]

**Hint:** Split the integral into two parts for easier integration.

### Step 4: Integrate
Now we can integrate each term:
\[ \frac{1}{3} \left( \int t^{1/2} \, dt - \int t^{-1/2} \, dt \right) \]
Calculating these integrals:
\[ \int t^{1/2} \, dt = \frac{t^{3/2}}{3/2} = \frac{2}{3} t^{3/2} \]
\[ \int t^{-1/2} \, dt = 2t^{1/2} \]
Thus, we have:
\[ \frac{1}{3} \left( \frac{2}{3} t^{3/2} - 2t^{1/2} \right) = \frac{2}{9} t^{3/2} - \frac{2}{3} t^{1/2} \]

**Hint:** Remember to include the constant of integration after integrating.

### Step 5: Substitute Back
Now substitute \( t = 1 + x^3 \) back into the expression:
\[ = \frac{2}{9} (1 + x^3)^{3/2} - \frac{2}{3} (1 + x^3)^{1/2} + C \]

**Hint:** Always revert back to the original variable to finalize your answer.

### Final Answer
Thus, the final answer is:
\[ \int \frac{x^5}{\sqrt{1 + x^3}} \, dx = \frac{2}{9} (1 + x^3)^{3/2} - \frac{2}{3} (1 + x^3)^{1/2} + C \]