If `int_(-1)^(3//2)|xsinpix|dx = (k)/(pi^(2))`, then the value of k is :

Answer

To solve the integral \( \int_{-1}^{\frac{3}{2}} |x \sin(\pi x)| \, dx \) and express it in the form \( \frac{k}{\pi^2} \), we will follow these steps:

### Step 1: Split the Integral
We need to identify the points where the expression inside the absolute value changes sign. The function \( x \sin(\pi x) \) changes sign at \( x = 0 \) and \( x = 1 \). Thus, we can split the integral into three parts:
\[ \int_{-1}^{\frac{3}{2}} |x \sin(\pi x)| \, dx = \int_{-1}^{0} |x \sin(\pi x)| \, dx + \int_{0}^{1} |x \sin(\pi x)| \, dx + \int_{1}^{\frac{3}{2}} |x \sin(\pi x)| \, dx \]

### Step 2: Evaluate Each Integral
1. **For \( x \in [-1, 0] \)**:
- Here, \( x \) is negative and \( \sin(\pi x) \) is also negative, thus \( |x \sin(\pi x)| = -x \sin(\pi x) \).
\[ \int_{-1}^{0} |x \sin(\pi x)| \, dx = \int_{-1}^{0} -x \sin(\pi x) \, dx \]

2. **For \( x \in [0, 1] \)**:
- Both \( x \) and \( \sin(\pi x) \) are positive, thus \( |x \sin(\pi x)| = x \sin(\pi x) \).
\[ \int_{0}^{1} |x \sin(\pi x)| \, dx = \int_{0}^{1} x \sin(\pi x) \, dx \]

3. **For \( x \in [1, \frac{3}{2}] \)**:
- Here, \( x \) is positive and \( \sin(\pi x) \) is negative, thus \( |x \sin(\pi x)| = -x \sin(\pi x) \).
\[ \int_{1}^{\frac{3}{2}} |x \sin(\pi x)| \, dx = \int_{1}^{\frac{3}{2}} -x \sin(\pi x) \, dx \]

### Step 3: Combine the Integrals
Now we can combine these integrals:
\[ \int_{-1}^{\frac{3}{2}} |x \sin(\pi x)| \, dx = -\int_{-1}^{0} x \sin(\pi x) \, dx + \int_{0}^{1} x \sin(\pi x) \, dx - \int_{1}^{\frac{3}{2}} x \sin(\pi x) \, dx \]

### Step 4: Evaluate the Integral \( \int x \sin(\pi x) \, dx \)
Using integration by parts:
Let \( u = x \) and \( dv = \sin(\pi x) \, dx \).
Then, \( du = dx \) and \( v = -\frac{1}{\pi} \cos(\pi x) \).

Applying integration by parts:
\[ \int x \sin(\pi x) \, dx = -\frac{x}{\pi} \cos(\pi x) + \frac{1}{\pi} \int \cos(\pi x) \, dx \]
\[ = -\frac{x}{\pi} \cos(\pi x) + \frac{1}{\pi^2} \sin(\pi x) \]

### Step 5: Evaluate the Limits
Now we need to evaluate the integral from the limits for each segment:
1. **From \(-1\) to \(0\)**:
\[ \left[-\frac{x}{\pi} \cos(\pi x) + \frac{1}{\pi^2} \sin(\pi x)\right]_{-1}^{0} \]

2. **From \(0\) to \(1\)**:
\[ \left[-\frac{x}{\pi} \cos(\pi x) + \frac{1}{\pi^2} \sin(\pi x)\right]_{0}^{1} \]

3. **From \(1\) to \(\frac{3}{2}\)**:
\[ \left[-\frac{x}{\pi} \cos(\pi x) + \frac{1}{\pi^2} \sin(\pi x)\right]_{1}^{\frac{3}{2}} \]

### Step 6: Combine All Results
After calculating each of these integrals and substituting the limits, we will sum them up to find the total value of the integral.

### Step 7: Find \( k \)
Finally, we will express the result in the form \( \frac{k}{\pi^2} \) and identify the value of \( k \).