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. Here’s a step-by-step solution:

### Step 1: Use the Trigonometric Identity
We know that:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
This identity allows us to express \( \sin^2 x \) in a form that is easier to integrate.

### Step 2: Substitute the Identity into the Integral
We can substitute this identity into the integral:
\[ \int \sin^2 x \, dx = \int \frac{1 - \cos(2x)}{2} \, dx \]
This simplifies to:
\[ \int \sin^2 x \, dx = \frac{1}{2} \int (1 - \cos(2x)) \, dx \]

### Step 3: Split the Integral
Now we can split the integral into two parts:
\[ \int \sin^2 x \, dx = \frac{1}{2} \left( \int 1 \, dx - \int \cos(2x) \, dx \right) \]

### Step 4: Integrate Each Part
Now we integrate each part separately:
1. The integral of \( 1 \) is simply \( x \).
2. The integral of \( \cos(2x) \) requires a substitution. The integral is:
\[ \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) \]
(We divide by 2 because of the chain rule.)

Putting this together, we have:
\[ \int \sin^2 x \, dx = \frac{1}{2} \left( x - \frac{1}{2} \sin(2x) \right) + C \]

### Step 5: Simplify the Expression
Now, simplifying the expression gives us:
\[ \int \sin^2 x \, dx = \frac{x}{2} - \frac{1}{4} \sin(2x) + C \]

### Final Result
Thus, the evaluated integral is:
\[ \int \sin^2 x \, dx = \frac{x}{2} - \frac{1}{4} \sin(2x) + C \]

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