`intx^5(1+x^3)^(2/3)dx`

Answer

To solve the integral \( \int x^5 (1+x^3)^{\frac{2}{3}} \, dx \), we will use substitution and integration techniques. Here’s a step-by-step solution:

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

### Step 2: Substitution
Let’s make the substitution:
\[ t = 1 + x^3 \]
Then, differentiating both sides gives:
\[ dt = 3x^2 \, dx \quad \Rightarrow \quad x^2 \, dx = \frac{dt}{3} \]
Also, from our substitution, we have:
\[ x^3 = t - 1 \]

### Step 3: Substitute in the Integral
Now we substitute \( x^3 \) and \( x^2 \, dx \) in the integral:
\[ \int (t-1) (t)^{\frac{2}{3}} \frac{dt}{3} \]
This simplifies to:
\[ \frac{1}{3} \int (t-1) t^{\frac{2}{3}} \, dt \]

### Step 4: Expand the Integral
Now we expand the integrand:
\[ \frac{1}{3} \int (t^{\frac{5}{3}} - t^{\frac{2}{3}}) \, dt \]

### Step 5: Integrate Term by Term
Now we integrate each term:
\[ \frac{1}{3} \left( \frac{t^{\frac{5}{3}+1}}{\frac{5}{3}+1} - \frac{t^{\frac{2}{3}+1}}{\frac{2}{3}+1} \right) + C \]
This simplifies to:
\[ \frac{1}{3} \left( \frac{t^{\frac{8}{3}}}{\frac{8}{3}} - \frac{t^{\frac{5}{3}}}{\frac{5}{3}} \right) + C \]
\[ = \frac{1}{3} \left( \frac{3}{8} t^{\frac{8}{3}} - \frac{3}{5} t^{\frac{5}{3}} \right) + C \]
\[ = \frac{1}{8} t^{\frac{8}{3}} - \frac{1}{5} t^{\frac{5}{3}} + C \]

### Step 6: Substitute Back
Finally, we substitute back \( t = 1 + x^3 \):
\[ = \frac{1}{8} (1 + x^3)^{\frac{8}{3}} - \frac{1}{5} (1 + x^3)^{\frac{5}{3}} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int x^5 (1+x^3)^{\frac{2}{3}} \, dx = \frac{1}{8} (1 + x^3)^{\frac{8}{3}} - \frac{1}{5} (1 + x^3)^{\frac{5}{3}} + C \]