To solve this definite integral, we can use the substitution method. Let $u = \sin\theta$. Then, the differential is: $du = \cos\theta \, d\theta$

Now, we change the limits of integration according to our substitution:

• When $\theta = 0$, $u = \sin(0) = 0$
• When $\theta = \pi/2$, $u = \sin(\pi/2) = 1$

Substituting these values into the integral: $\int_{0}^{1} u \, du = \left[ \frac{u^2}{2} \right]_{0}^{1}$ $\left[ \frac{1^2}{2} - \frac{0^2}{2} \right] = \frac{1}{2}$

Alternative Method: Using double-angle identity $\sin\theta \cos\theta = \frac{1}{2}\sin(2\theta)$: $\int_{0}^{\pi/2} \frac{1}{2}\sin(2\theta) \, d\theta = \frac{1}{2} \left[ -\frac{\cos(2\theta)}{2} \right]_{0}^{\pi/2} = -\frac{1}{4} [\cos(\pi) - \cos(0)] = -\frac{1}{4} [-1 - 1] = \frac{1}{2}$

• Option A is correct.
• Option B is incorrect as it neglects the factor of $1/2$ from the integration of $u$.
• Option C is incorrect and typically arises from a sign error in evaluating cosine values.
• Option D is incorrect and comes from double-counting the division factor in the trigonometric identity method.