The value of the integral `intx^((1)/(3))(1-sqrtx)^(3)dx` is equal to (where c is the constant of integration)

Options

A. `6((x^((4)/(3)))/(8)+(3)/(11)x^((11)/(6))+(3)/(14)x^((7)/(3))+(1)/(17)x^((17)/(6)))+c`
B. `6((x^((4)/(3)))/(8)-(3)/(11)x^((11)/(6))+(3)/(14)x^((7)/(3))-(1)/(17)x^((17)/(6)))+c`
C. `2((x^((4)/(3)))/(8)-(3)/(11)x^((11)/(6))-(3)/(14)x^((7)/(3))-(1)/(17)x^((17)/(6)))+c`
D. `2((x^(4))/(8)-(3)/(11)x^(11)-(3)/(11)x^(7)-(1)/(17)x^(17))+c`

Answer

To evaluate the integral \( I = \int x^{\frac{1}{3}} (1 - \sqrt{x})^3 \, dx \), we will follow these steps:

### Step 1: Expand the integrand
We start by expanding \( (1 - \sqrt{x})^3 \) using the binomial theorem:
\[ (1 - \sqrt{x})^3 = 1 - 3\sqrt{x} + 3x - x^{\frac{3}{2}} \]

### Step 2: Substitute the expanded form into the integral
Now, we substitute this expansion back into the integral:
\[ I = \int x^{\frac{1}{3}} \left( 1 - 3\sqrt{x} + 3x - x^{\frac{3}{2}} \right) \, dx \]

### Step 3: Distribute \( x^{\frac{1}{3}} \)
Distributing \( x^{\frac{1}{3}} \) across the terms gives us:
\[ I = \int \left( x^{\frac{1}{3}} - 3x^{\frac{1}{3}} \cdot \sqrt{x} + 3x^{\frac{4}{3}} - x^{\frac{3}{2}} \cdot x^{\frac{1}{3}} \right) \, dx \]
This simplifies to:
\[ I = \int \left( x^{\frac{1}{3}} - 3x^{\frac{5}{6}} + 3x^{\frac{4}{3}} - x^{\frac{5}{6}} \right) \, dx \]
Combining like terms:
\[ I = \int \left( x^{\frac{1}{3}} - 4x^{\frac{5}{6}} + 3x^{\frac{4}{3}} \right) \, dx \]

### Step 4: Integrate each term
Now we integrate each term separately:
1. \( \int x^{\frac{1}{3}} \, dx = \frac{x^{\frac{4}{3}}}{\frac{4}{3}} = \frac{3}{4} x^{\frac{4}{3}} \)
2. \( \int -4x^{\frac{5}{6}} \, dx = -4 \cdot \frac{x^{\frac{11}{6}}}{\frac{11}{6}} = -\frac{24}{11} x^{\frac{11}{6}} \)
3. \( \int 3x^{\frac{4}{3}} \, dx = 3 \cdot \frac{x^{\frac{7}{3}}}{\frac{7}{3}} = \frac{9}{7} x^{\frac{7}{3}} \)

### Step 5: Combine the results
Putting it all together, we have:
\[ I = \frac{3}{4} x^{\frac{4}{3}} - \frac{24}{11} x^{\frac{11}{6}} + \frac{9}{7} x^{\frac{7}{3}} + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the value of the integral is:
\[ I = \frac{3}{4} x^{\frac{4}{3}} - \frac{24}{11} x^{\frac{11}{6}} + \frac{9}{7} x^{\frac{7}{3}} + C \]

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