`int_(pi//2)^(0)sin^(11)xdx`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \sin^{11} x \, dx \), we will use the properties of definite integrals and the reduction formula for sine integrals.

### Step 1: Change the limits of integration
Using the property of definite integrals, if we change the limits of integration, we have:
\[ I = \int_{0}^{\frac{\pi}{2}} \sin^{11} x \, dx = -\int_{\frac{\pi}{2}}^{0} \sin^{11} x \, dx \]
This means that if we reverse the limits, the integral changes its sign.

### Step 2: Apply the reduction formula
The reduction formula for the integral of sine raised to an odd power is given by:
\[ \int \sin^n x \, dx = -\frac{1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x \, dx \]
For our case, we will use:
\[ \int_{0}^{\frac{\pi}{2}} \sin^{11} x \, dx = \frac{10}{11} \int_{0}^{\frac{\pi}{2}} \sin^{9} x \, dx \]

### Step 3: Continue applying the reduction formula
We will continue applying the reduction formula:
\[ \int_{0}^{\frac{\pi}{2}} \sin^{9} x \, dx = \frac{8}{9} \int_{0}^{\frac{\pi}{2}} \sin^{7} x \, dx \]
\[ \int_{0}^{\frac{\pi}{2}} \sin^{7} x \, dx = \frac{6}{7} \int_{0}^{\frac{\pi}{2}} \sin^{5} x \, dx \]
\[ \int_{0}^{\frac{\pi}{2}} \sin^{5} x \, dx = \frac{4}{5} \int_{0}^{\frac{\pi}{2}} \sin^{3} x \, dx \]
\[ \int_{0}^{\frac{\pi}{2}} \sin^{3} x \, dx = \frac{2}{3} \int_{0}^{\frac{\pi}{2}} \sin^{1} x \, dx \]
\[ \int_{0}^{\frac{\pi}{2}} \sin^{1} x \, dx = 1 \]

### Step 4: Substitute back to find the integral
Now we can substitute back to find \( I \):
\[ \int_{0}^{\frac{\pi}{2}} \sin^{3} x \, dx = \frac{2}{3} \cdot 1 = \frac{2}{3} \]
\[ \int_{0}^{\frac{\pi}{2}} \sin^{5} x \, dx = \frac{4}{5} \cdot \frac{2}{3} = \frac{8}{15} \]
\[ \int_{0}^{\frac{\pi}{2}} \sin^{7} x \, dx = \frac{6}{7} \cdot \frac{8}{15} = \frac{48}{105} \]
\[ \int_{0}^{\frac{\pi}{2}} \sin^{9} x \, dx = \frac{8}{9} \cdot \frac{48}{105} = \frac{384}{945} \]
\[ I = \frac{10}{11} \cdot \frac{384}{945} = \frac{3840}{10395} \]

### Final Answer
Thus, the value of the integral \( I = \int_{0}^{\frac{\pi}{2}} \sin^{11} x \, dx \) is:
\[ \boxed{\frac{3840}{10395}} \]