We need to evaluate the integral of sine x times cosine x. This is a classic integral that can be solved using different methods. Let's visualize the function first - it's the product of sine and cosine, which creates this wave pattern.
The first method uses the double-angle identity. Since sine of 2x equals 2 sine x cosine x, we can rewrite sine x cosine x as one-half sine 2x. This transforms our integral into a simpler form that's easier to evaluate.
The second method uses u-substitution. We let u equal sine x, so du equals cosine x dx. This transforms our integral into the integral of u du, which equals u squared over 2 plus C. Substituting back gives us sine squared x over 2 plus C.
Now let's complete the first method. We integrate sine 2x to get negative one-half cosine 2x. Multiplying by one-half gives us negative one-fourth cosine 2x plus C. This is our final answer using the double-angle identity approach.
Both methods give equivalent results that differ only by a constant. Using the identity cosine 2x equals 1 minus 2 sine squared x, we can show that negative one-fourth cosine 2x equals negative one-fourth plus sine squared x over 2. Since the constant of integration can absorb the negative one-fourth term, both forms are valid answers.