`"If "int x^(5)(1+x^(3))^(2//3)dx=A(1+x^(3))^(8//3)+B(1+x^(3))^(5//3)+c, " then " `

Options

A. `A=(1)/(4),B=(1)/(5)`
B. `A=(1)/(8),B=-(1)/(5)`
C. `A=-(1)/(8),B=(1)/(5)`
D. non of these

Explanation

To solve the integral \( \int x^5 (1 + x^3)^{\frac{2}{3}} \, dx \) and express it in the form \( A(1 + x^3)^{\frac{8}{3}} + B(1 + x^3)^{\frac{5}{3}} + C \), we will follow these steps:

### Step 1: Substitution
Let \( t = 1 + x^3 \). Then, we differentiate to find \( dx \):
\[ dt = 3x^2 \, dx \implies dx = \frac{dt}{3x^2} \]
Also, we can express \( x^2 \) in terms of \( t \):
\[ x^3 = t - 1 \implies x = (t - 1)^{\frac{1}{3}} \implies x^2 = (t - 1)^{\frac{2}{3}} \]

### Step 2: Rewrite the Integral
Now, substituting \( x^2 \) and \( dx \) into the integral:
\[ \int x^5 (1 + x^3)^{\frac{2}{3}} \, dx = \int (t - 1)^{\frac{5}{3}} t^{\frac{2}{3}} \cdot \frac{dt}{3(t - 1)^{\frac{2}{3}}} \]
This simplifies to:
\[ \frac{1}{3} \int (t - 1)^{\frac{5}{3} - \frac{2}{3}} t^{\frac{2}{3}} \, dt = \frac{1}{3} \int (t - 1)^{1} t^{\frac{2}{3}} \, dt \]

### Step 3: Expand the Integral
Now, expand \( (t - 1) t^{\frac{2}{3}} \):
\[ (t - 1) t^{\frac{2}{3}} = t^{\frac{5}{3}} - t^{\frac{2}{3}} \]
Thus, the integral becomes:
\[ \frac{1}{3} \int (t^{\frac{5}{3}} - t^{\frac{2}{3}}) \, dt = \frac{1}{3} \left( \frac{3}{8} t^{\frac{8}{3}} - \frac{3}{5} t^{\frac{5}{3}} \right) + C \]

### Step 4: Simplify the Integral
Simplifying gives:
\[ \frac{1}{8} t^{\frac{8}{3}} - \frac{1}{5} t^{\frac{5}{3}} + C \]

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

### Step 6: Identify Constants
From the expression, we can identify:
- \( A = \frac{1}{8} \)
- \( B = -\frac{1}{5} \)
- \( C = C \)

### Final Answer
Thus, we have:
\[ \int x^5 (1 + x^3)^{\frac{2}{3}} \, dx = \frac{1}{8} (1 + x^3)^{\frac{8}{3}} - \frac{1}{5} (1 + x^3)^{\frac{5}{3}} + C \]