If `int_(pi//2)^(theta) sin x dx=sin 2 theta` then the of `theta` satisfying `0 lt theta lt pi`, is

Options

A. `3pi//2`
B. `pi//6`
C. `5pi//6`
D. `pi//2`

Explanation

To solve the problem, we need to evaluate the integral and set it equal to \( \sin 2\theta \) as given in the question. Let's go through the steps systematically.

### Step 1: Set Up the Integral
We start with the integral:
\[ I = \int_{\frac{\pi}{2}}^{\theta} \sin x \, dx \]

### Step 2: Evaluate the Integral
The integral of \( \sin x \) is:
\[ \int \sin x \, dx = -\cos x \]
Now we can evaluate the definite integral from \( \frac{\pi}{2} \) to \( \theta \):
\[ I = \left[-\cos x\right]_{\frac{\pi}{2}}^{\theta} = -\cos(\theta) - \left(-\cos\left(\frac{\pi}{2}\right)\right) \]
Since \( \cos\left(\frac{\pi}{2}\right) = 0 \), we have:
\[ I = -\cos(\theta) + 0 = -\cos(\theta) \]

### Step 3: Set Up the Equation
According to the problem, this integral equals \( \sin 2\theta \):
\[ -\cos(\theta) = \sin 2\theta \]
Using the double angle identity for sine, we can rewrite \( \sin 2\theta \):
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Thus, we have:
\[ -\cos(\theta) = 2 \sin \theta \cos \theta \]

### Step 4: Rearranging the Equation
Rearranging gives us:
\[ \cos(\theta) + 2 \sin \theta \cos \theta = 0 \]
Factoring out \( \cos(\theta) \):
\[ \cos(\theta)(1 + 2 \sin \theta) = 0 \]

### Step 5: Solve for \( \theta \)
This equation gives us two cases to consider:
1. \( \cos(\theta) = 0 \)
2. \( 1 + 2 \sin \theta = 0 \)

#### Case 1: \( \cos(\theta) = 0 \)
The solutions to \( \cos(\theta) = 0 \) in the interval \( (0, \pi) \) is:
\[ \theta = \frac{\pi}{2} \]

#### Case 2: \( 1 + 2 \sin \theta = 0 \)
Solving for \( \sin \theta \):
\[ 2 \sin \theta = -1 \quad \Rightarrow \quad \sin \theta = -\frac{1}{2} \]
However, \( \sin \theta \) is negative in the interval \( (0, \pi) \) only at \( \theta = \frac{7\pi}{6} \) and \( \theta = \frac{11\pi}{6} \), which are outside the given range \( (0, \pi) \).

### Conclusion
The only solution for \( \theta \) in the interval \( (0, \pi) \) is:
\[ \theta = \frac{\pi}{2} \]

### Final Answer
Thus, the value of \( \theta \) satisfying the condition is:
\[ \theta = \frac{\pi}{2} \]