The antiderivative of
sin^2(x) can be found using a trigonometric identity and integration techniques. To start, we use the identity for
sin^2(x), which is expressed in terms of cosine:
\[ sin^2(x) = \frac{1 – cos(2x)}{2} \]
This reformulation allows us to express the integral in a simpler form:
\[ \int sin^2(x) \, dx = \int \frac{1 – cos(2x)}{2} \, dx \]
Now, we can separate the integral:
\[ \int sin^2(x) \, dx = \frac{1}{2} \int 1 \, dx – \frac{1}{2} \int cos(2x) \, dx \]
Integrating each term gives:
\[ \frac{1}{2}(x) – \frac{1}{2}\left(\frac{sin(2x)}{2}\right) + C \]
Combining the results, we find the final answer for the antiderivative:
\[ \ rac{x}{2} – \frac{sin(2x)}{4} + C \]
where C is the constant of integration. Thus, the antiderivative of
sin^2(x) is:
\[ \int sin^2(x) \, dx = \frac{x}{2} – \frac{sin(2x)}{4} + C \]