`IfI_n=int_0^1(1-x^5)^n dx ,t h e n(55)/7(I_(10))/(I_(11))` is equal to

Explanation

To solve the problem, we need to evaluate the expression \( \frac{55}{7} \cdot \frac{I_{10}}{I_{11}} \), where \( I_n = \int_0^1 (1 - x^5)^n \, dx \).

### Step 1: Define the integral
We start with the definition of \( I_n \):
\[ I_n = \int_0^1 (1 - x^5)^n \, dx \]

### Step 2: Find \( I_{11} \) using integration by parts
To find \( I_{11} \), we can use integration by parts. We set:
- \( u = (1 - x^5)^{11} \) (first function)
- \( dv = dx \) (second function)

Then, we differentiate and integrate:
- \( du = -55x^4(1 - x^5)^{10} \, dx \)
- \( v = x \)

Using integration by parts:
\[ I_{11} = \left[ (1 - x^5)^{11} \cdot x \right]_0^1 - \int_0^1 x \cdot (-55x^4(1 - x^5)^{10}) \, dx \]

Evaluating the boundary terms:
\[ \left[ (1 - x^5)^{11} \cdot x \right]_0^1 = (1 - 1)^{11} \cdot 1 - (1 - 0)^{11} \cdot 0 = 0 - 0 = 0 \]

Thus, we have:
\[ I_{11} = 55 \int_0^1 x^5 (1 - x^5)^{10} \, dx \]

### Step 3: Change the variable in the integral
Now, we can express \( I_{11} \) in terms of \( I_{10} \):
\[ I_{11} = 55 \int_0^1 x^5 (1 - x^5)^{10} \, dx \]

Using the substitution \( x^5 = t \), we have \( dx = \frac{1}{5} t^{-\frac{4}{5}} dt \) and the limits change from \( x = 0 \) to \( x = 1 \) (which corresponds to \( t = 0 \) to \( t = 1 \)):
\[ I_{11} = 55 \cdot \frac{1}{5} \int_0^1 (1 - t)^{10} t^{\frac{1}{5} - 1} \, dt \]
This integral is a Beta function and can be expressed as:
\[ I_{11} = 11 \cdot B\left(\frac{6}{5}, 11\right) \]

### Step 4: Relate \( I_{10} \) and \( I_{11} \)
Using a similar process, we find \( I_{10} \):
\[ I_{10} = 55 \cdot \frac{1}{5} \int_0^1 (1 - t)^{9} t^{\frac{1}{5} - 1} \, dt \]
Thus, we have:
\[ \frac{I_{10}}{I_{11}} = \frac{B\left(\frac{6}{5}, 10\right)}{B\left(\frac{6}{5}, 11\right)} = \frac{10}{11} \]

### Step 5: Substitute back into the expression
Now substituting back into the original expression:
\[ \frac{I_{10}}{I_{11}} = \frac{10}{11} \]

### Step 6: Calculate the final result
Now we can calculate:
\[ \frac{55}{7} \cdot \frac{I_{10}}{I_{11}} = \frac{55}{7} \cdot \frac{10}{11} = \frac{50}{7} \]

Thus, the final answer is:
\[ \boxed{8} \]