`"Let " k(x)=int((x^(2)+1)dx)/(root(3)(x^(3)+3x+6)) " and " k(-1)=(1)/(root(3)(2)). " Then the value of " k(-2) " is "-.`

Explanation

To solve the problem step by step, we start with the function defined as:

\[ k(x) = \frac{\int (x^2 + 1) \, dx}{\sqrt[3]{x^3 + 3x + 6}} \]

We are given that \( k(-1) = \frac{1}{\sqrt[3]{2}} \) and we need to find \( k(-2) \).

### Step 1: Substitute \( T \)
Let \( T = x^3 + 3x + 6 \). Then, we differentiate \( T \):

\[ \frac{dT}{dx} = 3x^2 + 3 \]

This implies:

\[ dT = (3x^2 + 3) \, dx \]

### Step 2: Rearranging \( dx \)
From the above, we can express \( dx \):

\[ dx = \frac{dT}{3(x^2 + 1)} \]

### Step 3: Substitute in the Integral
Now substitute \( dx \) into the integral:

\[ k(x) = \frac{\int (x^2 + 1) \, dx}{\sqrt[3]{T}} \]

Substituting for \( dx \):

\[ k(x) = \frac{\int (x^2 + 1) \cdot \frac{dT}{3(x^2 + 1)}}{\sqrt[3]{T}} \]

The \( (x^2 + 1) \) cancels out:

\[ k(x) = \frac{1}{3} \int dT \cdot \frac{1}{\sqrt[3]{T}} \]

### Step 4: Evaluate the Integral
The integral simplifies to:

\[ k(x) = \frac{1}{3} \cdot \frac{T^{2/3}}{2/3} + C = \frac{T^{2/3}}{2} + C \]

### Step 5: Substitute Back for \( T \)
Now substitute back \( T = x^3 + 3x + 6 \):

\[ k(x) = \frac{(x^3 + 3x + 6)^{2/3}}{2} + C \]

### Step 6: Use Given Information to Find \( C \)
We know \( k(-1) = \frac{1}{\sqrt[3]{2}} \):

Substituting \( x = -1 \):

\[ k(-1) = \frac{((-1)^3 + 3(-1) + 6)^{2/3}}{2} + C \]

Calculating \( (-1)^3 + 3(-1) + 6 = -1 - 3 + 6 = 2 \):

\[ k(-1) = \frac{(2)^{2/3}}{2} + C = \frac{2^{2/3}}{2} + C \]

Setting this equal to \( \frac{1}{\sqrt[3]{2}} \):

\[ \frac{2^{2/3}}{2} + C = \frac{1}{2^{1/3}} \]

### Step 7: Solve for \( C \)
To solve for \( C \):

\[ C = \frac{1}{2^{1/3}} - \frac{2^{2/3}}{2} \]

### Step 8: Calculate \( k(-2) \)
Now we need to find \( k(-2) \):

Substituting \( x = -2 \):

\[ k(-2) = \frac{((-2)^3 + 3(-2) + 6)^{2/3}}{2} + C \]

Calculating \( (-2)^3 + 3(-2) + 6 = -8 - 6 + 6 = -8 \):

\[ k(-2) = \frac{(-8)^{2/3}}{2} + C \]

Calculating \( (-8)^{2/3} = (8^{2/3}) = 4 \):

\[ k(-2) = \frac{4}{2} + C = 2 + C \]

### Step 9: Substitute \( C \)
Now substitute \( C \) back into the equation:

\[ k(-2) = 2 + \left( \frac{1}{2^{1/3}} - \frac{2^{2/3}}{2} \right) \]

### Final Calculation
Calculating the final value:

\[ k(-2) = 2 + \frac{1}{2^{1/3}} - \frac{2^{2/3}}{2} \]

This gives us the final value of \( k(-2) \).