We can sometimes use trigonometric identities to transform integrals. The integral formulas for `sin^(2)x and cos^(2)x` arise frequently in applications. Evaluate:
`intsin^(2)xdx`

Explanation

To evaluate the integral of \( \sin^2 x \, dx \), we can use a trigonometric identity to simplify the expression. The relevant identity is:

\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]

### Step 1: Substitute the Identity
We start by substituting the identity into the integral:

\[ \int \sin^2 x \, dx = \int \frac{1 - \cos(2x)}{2} \, dx \]

### Step 2: Simplify the Integral
Now, we can simplify the integral:

\[ \int \sin^2 x \, dx = \frac{1}{2} \int (1 - \cos(2x)) \, dx \]

### Step 3: Split the Integral
Next, we can split the integral into two separate integrals:

\[ \int \sin^2 x \, dx = \frac{1}{2} \left( \int 1 \, dx - \int \cos(2x) \, dx \right) \]

### Step 4: Evaluate Each Integral
Now we evaluate each integral separately:

1. The integral of \( 1 \, dx \) is simply \( x \).
2. The integral of \( \cos(2x) \, dx \) can be evaluated using the substitution method. The integral is:

\[ \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) \]

### Step 5: Combine the Results
Now we can combine the results from the two integrals:

\[ \int \sin^2 x \, dx = \frac{1}{2} \left( x - \frac{1}{2} \sin(2x) \right) + C \]

### Step 6: Simplify the Expression
Finally, we can simplify the expression:

\[ \int \sin^2 x \, dx = \frac{x}{2} - \frac{1}{4} \sin(2x) + C \]

### Final Answer
Thus, the evaluated integral is:

\[ \int \sin^2 x \, dx = \frac{x}{2} - \frac{1}{4} \sin(2x) + C \]