`int (x)^(1/3) (root(7)(1+root(3)(x^(4))))dx` is equal to

Options

A. `(21)/(32){1+root(3)(x^(4))}^(8//7)+C`
B. `(32)/(21){1+root(3)(x^(4))}^(8//7)+C`
C. `(7)/(32){1+root(3)(x^(4))}^(8//7)+C`
D. none of these

Answer

To solve the integral \( I = \int x^{1/3} \left(1 + (x^{4/3})\right)^{1/7} dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = 1 + x^{4/3} \). Then, we differentiate both sides to find \( dx \) in terms of \( dt \).

\[ \frac{dt}{dx} = \frac{4}{3} x^{1/3} \implies dt = \frac{4}{3} x^{1/3} dx \implies dx = \frac{3}{4} \frac{dt}{x^{1/3}} \]

### Step 2: Express \( x^{1/3} \) in terms of \( t \)
From our substitution \( t = 1 + x^{4/3} \), we can express \( x^{4/3} \) as \( t - 1 \). Thus,

\[ x^{4/3} = t - 1 \implies x^{1/3} = (t - 1)^{3/4} \]

### Step 3: Substitute back into the integral
Now we can substitute \( x^{1/3} \) and \( dx \) into the integral:

\[ I = \int (t - 1)^{3/4} t^{1/7} \cdot \frac{3}{4} \frac{dt}{(t - 1)^{3/4}} = \frac{3}{4} \int t^{1/7} dt \]

### Step 4: Integrate
Now we can integrate:

\[ I = \frac{3}{4} \cdot \frac{t^{1/7 + 1}}{1/7 + 1} + C = \frac{3}{4} \cdot \frac{t^{8/7}}{8/7} + C = \frac{3}{4} \cdot \frac{7}{8} t^{8/7} + C \]

### Step 5: Substitute back for \( t \)
Now substitute back \( t = 1 + x^{4/3} \):

\[ I = \frac{21}{32} (1 + x^{4/3})^{8/7} + C \]

### Final Answer
Thus, the integral evaluates to:

\[ I = \frac{21}{32} (1 + x^{4/3})^{8/7} + C \]

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