Evaluate :
`int_(0)^(pi)xsin^(5)xdx`

Answer

To evaluate the integral \( I = \int_{0}^{\pi} x \sin^5 x \, dx \), we can use a property of definite integrals. Let's go through the steps:

### Step 1: Set up the integral
Let
\[ I = \int_{0}^{\pi} x \sin^5 x \, dx \]

### Step 2: Use the property of definite integrals
Using the property that states:
\[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx \]
we can rewrite the integral by substituting \( x \) with \( \pi - x \):
\[ I = \int_{0}^{\pi} (\pi - x) \sin^5(\pi - x) \, dx \]

### Step 3: Simplify the integral
Since \( \sin(\pi - x) = \sin x \), we can rewrite the integral:
\[ I = \int_{0}^{\pi} (\pi - x) \sin^5 x \, dx \]
This expands to:
\[ I = \int_{0}^{\pi} \pi \sin^5 x \, dx - \int_{0}^{\pi} x \sin^5 x \, dx \]
Thus, we have:
\[ I = \pi \int_{0}^{\pi} \sin^5 x \, dx - I \]

### Step 4: Solve for \( I \)
Rearranging gives:
\[ 2I = \pi \int_{0}^{\pi} \sin^5 x \, dx \]
So,
\[ I = \frac{\pi}{2} \int_{0}^{\pi} \sin^5 x \, dx \]

### Step 5: Evaluate \( \int_{0}^{\pi} \sin^5 x \, dx \)
To evaluate \( \int_{0}^{\pi} \sin^5 x \, dx \), we can use the reduction formula or the known result:
\[ \int_{0}^{\pi} \sin^n x \, dx = \frac{(n-1)!!}{n!!} \cdot \pi \]
For \( n = 5 \):
\[ \int_{0}^{\pi} \sin^5 x \, dx = \frac{4!!}{5!!} \cdot \pi = \frac{8}{15} \cdot \pi \]

### Step 6: Substitute back to find \( I \)
Now substituting back into our equation for \( I \):
\[ I = \frac{\pi}{2} \cdot \frac{8}{15} \cdot \pi = \frac{8\pi^2}{30} = \frac{4\pi^2}{15} \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{\frac{4\pi^2}{15}} \]