Step 1: Understanding the Concept:
We cannot directly integrate $\sin^2 x$. The standard technique is to use a trigonometric identity to reduce the power of the function to one, making it easily integrable.
Step 2: Key Formula or Approach:
We use the power-reduction formula for sine, which is derived from the double-angle identity for cosine:
\[ \cos(2x) = 1 - 2\sin^2 x \]
Rearranging this formula to solve for $\sin^2 x$, we get:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
Step 3: Detailed Explanation:
The integral is $\int \sin^2 x \,dx$.
Substitute the power-reduction formula into the integral:
\[ \int \frac{1 - \cos(2x)}{2} \,dx \]
Split the integral into two parts:
\[ \int \left(\frac{1}{2} - \frac{\cos(2x)}{2}\right) \,dx = \int \frac{1}{2} \,dx - \frac{1}{2} \int \cos(2x) \,dx \]
Now integrate each part:
\[ \int \frac{1}{2} \,dx = \frac{1}{2}x \]
For the second part, $\int \cos(2x) \,dx$, we get $\frac{\sin(2x)}{2}$. So:
\[ -\frac{1}{2} \int \cos(2x) \,dx = -\frac{1}{2} \left(\frac{\sin(2x)}{2}\right) = -\frac{\sin(2x)}{4} \]
Combine the parts and add the constant of integration, C:
\[ \frac{1}{2}x - \frac{\sin(2x)}{4} + C \]
Step 4: Final Answer:
The result of the integration is $\frac{x}{2} - \frac{\sin 2x}{4} + c$. Therefore, option (D) is correct.