The value of `int_(0)^(pi) x sin^(3) x dx ` is

Options

A. `(4pi)/3`
B. `(2pi)/3`
C. 0
D. None of these

Explanation

To solve the integral \( I = \int_{0}^{\pi} x \sin^3 x \, dx \), we can use the property of definite integrals that states:

\[ \int_{0}^{A} f(x) \, dx = \int_{0}^{A} f(A - x) \, dx \]

### Step 1: Set up the integral
Let \( I = \int_{0}^{\pi} x \sin^3 x \, dx \).

### Step 2: Use the property of definite integrals
We can express \( I \) in another form by substituting \( x \) with \( \pi - x \):

\[ I = \int_{0}^{\pi} (\pi - x) \sin^3(\pi - x) \, dx \]

Since \( \sin(\pi - x) = \sin x \), we can rewrite this as:

\[ I = \int_{0}^{\pi} (\pi - x) \sin^3 x \, dx \]

### Step 3: Expand the integral
Now, we can expand this integral:

\[ I = \int_{0}^{\pi} \pi \sin^3 x \, dx - \int_{0}^{\pi} x \sin^3 x \, dx \]

This gives us:

\[ I = \pi \int_{0}^{\pi} \sin^3 x \, dx - I \]

### Step 4: Solve for \( I \)
Now, we can add \( I \) to both sides:

\[ 2I = \pi \int_{0}^{\pi} \sin^3 x \, dx \]

Thus, we have:

\[ I = \frac{\pi}{2} \int_{0}^{\pi} \sin^3 x \, dx \]

### Step 5: Calculate \( \int_{0}^{\pi} \sin^3 x \, dx \)
To compute \( \int_{0}^{\pi} \sin^3 x \, dx \), we can use the identity:

\[ \sin^3 x = \frac{3\sin x - \sin(3x)}{4} \]

Thus,

\[ \int_{0}^{\pi} \sin^3 x \, dx = \int_{0}^{\pi} \frac{3\sin x - \sin(3x)}{4} \, dx \]

### Step 6: Evaluate the integrals
Now we can evaluate the integral:

\[ \int_{0}^{\pi} \sin x \, dx = 2 \]
\[ \int_{0}^{\pi} \sin(3x) \, dx = 0 \]

So,

\[ \int_{0}^{\pi} \sin^3 x \, dx = \frac{1}{4} \left( 3 \cdot 2 - 0 \right) = \frac{3}{2} \]

### Step 7: Substitute back into the equation for \( I \)
Now substituting back, we get:

\[ I = \frac{\pi}{2} \cdot \frac{3}{2} = \frac{3\pi}{4} \]

### Step 8: Final result
Thus, the value of the integral \( \int_{0}^{\pi} x \sin^3 x \, dx \) is:

\[ \boxed{\frac{3\pi}{4}} \]