`int_(-pi//4)^(pi//4)sin^(2)x dx=`

Options

A. `(pi-4)/(8)`
B. `(pi-2)/(4)`
C. `(pi-2)/(2)`
D. `pi`

Answer

To solve the integral \( \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin^2 x \, dx \), we can follow these steps:

### Step 1: Use the identity for \(\sin^2 x\)
We know that:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
Thus, we can rewrite the integral as:
\[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin^2 x \, dx = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1 - \cos(2x)}{2} \, dx \]

### Step 2: Factor out the constant
Since \(\frac{1}{2}\) is a constant, we can factor it out of the integral:
\[ = \frac{1}{2} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (1 - \cos(2x)) \, dx \]

### Step 3: Split the integral
Now, we can split the integral into two parts:
\[ = \frac{1}{2} \left( \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} 1 \, dx - \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos(2x) \, dx \right) \]

### Step 4: Evaluate the first integral
The first integral, \(\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} 1 \, dx\), is simply the length of the interval:
\[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} 1 \, dx = \left[ x \right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} = \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2} \]

### Step 5: Evaluate the second integral
Now, we evaluate the second integral:
\[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos(2x) \, dx \]
The integral of \(\cos(2x)\) is \(\frac{1}{2} \sin(2x)\):
\[ \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) \]
Thus, we have:
\[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos(2x) \, dx = \left[ \frac{1}{2} \sin(2x) \right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} = \frac{1}{2} \left( \sin\left(\frac{\pi}{2}\right) - \sin\left(-\frac{\pi}{2}\right) \right) = \frac{1}{2} (1 - (-1)) = \frac{1}{2} \cdot 2 = 1 \]

### Step 6: Combine the results
Now, we can combine the results:
\[ = \frac{1}{2} \left( \frac{\pi}{2} - 1 \right) = \frac{\pi}{4} - \frac{1}{2} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin^2 x \, dx = \frac{\pi}{4} - \frac{1}{2} \]