If `I = int_(0)^(2pi)sin^(2)xdx`, then

Options

A. A.`I = 2 underset(0)overset(pi)intsin^(2)xdx`
B. B.`I = 4 underset(0)overset(pi//2)intsin^(2)xdx`
C. C.`I = underset(0)overset(2pi)intcos^(2)xdx`
D. D.`I = 8 underset(0)overset(pi//4)intsin^(2)xdx`

Answer

To solve the integral \( I = \int_{0}^{2\pi} \sin^2 x \, dx \), we can use the properties of definite integrals and some trigonometric identities. Here’s the step-by-step solution:

### Step 1: Use the property of definite integrals
We can use the property of definite integrals that states:
\[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx \]
In our case, we can express the integral from \(0\) to \(2\pi\) as:
\[ I = \int_{0}^{2\pi} \sin^2 x \, dx = \int_{0}^{2\pi} \sin^2(2\pi - x) \, dx \]
Since \(\sin(2\pi - x) = -\sin x\), we have:
\[ \sin^2(2\pi - x) = \sin^2 x \]
Thus, the integral remains unchanged.

### Step 2: Split the integral
We can split the integral from \(0\) to \(2\pi\) into two parts:
\[ I = \int_{0}^{\pi} \sin^2 x \, dx + \int_{\pi}^{2\pi} \sin^2 x \, dx \]
Using the substitution \(x = 2\pi - t\) for the second integral:
\[ \int_{\pi}^{2\pi} \sin^2 x \, dx = \int_{0}^{\pi} \sin^2(2\pi - t) \, (-dt) = \int_{0}^{\pi} \sin^2 t \, dt \]
So we can write:
\[ I = 2 \int_{0}^{\pi} \sin^2 x \, dx \]

### Step 3: Evaluate \(\int_{0}^{\pi} \sin^2 x \, dx\)
To evaluate \(\int_{0}^{\pi} \sin^2 x \, dx\), we can use the identity:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
Thus,
\[ \int_{0}^{\pi} \sin^2 x \, dx = \int_{0}^{\pi} \frac{1 - \cos(2x)}{2} \, dx \]
This can be split into two integrals:
\[ = \frac{1}{2} \int_{0}^{\pi} 1 \, dx - \frac{1}{2} \int_{0}^{\pi} \cos(2x) \, dx \]

### Step 4: Calculate the integrals
Calculating the first integral:
\[ \int_{0}^{\pi} 1 \, dx = \pi \]
Calculating the second integral:
\[ \int_{0}^{\pi} \cos(2x) \, dx = \left[ \frac{\sin(2x)}{2} \right]_{0}^{\pi} = \frac{\sin(2\pi)}{2} - \frac{\sin(0)}{2} = 0 \]
Putting it all together:
\[ \int_{0}^{\pi} \sin^2 x \, dx = \frac{1}{2} \left( \pi - 0 \right) = \frac{\pi}{2} \]

### Step 5: Final calculation of \(I\)
Now substituting back into our expression for \(I\):
\[ I = 2 \int_{0}^{\pi} \sin^2 x \, dx = 2 \cdot \frac{\pi}{2} = \pi \]

### Final Answer
Thus, the value of the integral is:
\[ I = \pi \]