`int(x^(2)+x)(x^(-8)+2x^(-9))^(1//10)dx` is equal to

Options

A. `(5)/(11)(x^(2)+2x)^(11//10)+C`
B. `(5)/(11)(x+1)^(11//10)+C`
C. `(6)/(7)(x+1)^(11//10)+C`
D. `(11)/(5)(x^(2)+2x)^(11//10)+C`

Explanation

To solve the integral \( \int (x^2 + x)(x^{-8} + 2x^{-9})^{\frac{1}{10}} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int (x^2 + x)(x^{-8} + 2x^{-9})^{\frac{1}{10}} \, dx \]

### Step 2: Factor Out \( x \)
We can factor \( x \) from \( x^2 + x \):
\[ = \int x(x + 1)(x^{-8} + 2x^{-9})^{\frac{1}{10}} \, dx \]

### Step 3: Simplify the Expression Inside the Integral
Now, we simplify the expression inside the integral:
\[ = \int x(x + 1)(x^{-8} + 2x^{-9})^{\frac{1}{10}} \, dx \]
This can be rewritten as:
\[ = \int x(x + 1)(x^{-8} + 2x^{-9})^{\frac{1}{10}} \, dx \]

### Step 4: Substitute \( t = x^2 + 2x \)
Let’s make a substitution. Let:
\[ t = x^2 + 2x \]
Then, differentiate \( t \):
\[ dt = (2x + 2) \, dx = 2(x + 1) \, dx \quad \Rightarrow \quad dx = \frac{dt}{2(x + 1)} \]

### Step 5: Substitute Back into the Integral
Substituting \( t \) and \( dx \) back into the integral:
\[ = \int x \cdot t^{\frac{1}{10}} \cdot \frac{dt}{2(x + 1)} \]
Notice that \( x = \frac{t - 2x}{2} \). However, we can simplify this further by recognizing that:
\[ = \frac{1}{2} \int t^{\frac{1}{10}} \, dt \]

### Step 6: Integrate
Now we can integrate:
\[ = \frac{1}{2} \cdot \frac{t^{\frac{1}{10} + 1}}{\frac{1}{10} + 1} + C \]
\[ = \frac{1}{2} \cdot \frac{t^{\frac{11}{10}}}{\frac{11}{10}} + C \]
\[ = \frac{5}{11} t^{\frac{11}{10}} + C \]

### Step 7: Substitute Back for \( t \)
Now we substitute back \( t = x^2 + 2x \):
\[ = \frac{5}{11} (x^2 + 2x)^{\frac{11}{10}} + C \]

### Final Answer
Thus, the final answer is:
\[ \int (x^2 + x)(x^{-8} + 2x^{-9})^{\frac{1}{10}} \, dx = \frac{5}{11} (x^2 + 2x)^{\frac{11}{10}} + C \]