Using the reduction formula can you please find sin^2(x)dx

To find the integral of \( \sin^2(x) \, dx \) using a reduction formula, we can use the identity:

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

This transforms the integral:

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

We can separate this into two integrals:

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

Now, we can evaluate each part:

1. The first integral is straightforward:

\[
\int 1 \, dx = x
\]

2. For the second integral, we can use the fact that:

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

Putting it all together:

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

This simplifies to:

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

where \( C \) is the constant of integration.

### Summary
Thus, the integral of \( \sin^2(x) \, dx \) is:

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

Would you like to explore more details on this process, or do you have any questions? Here are some related questions you might consider:

1. What are other methods to integrate \( \sin^2(x) \)?
2. Can you explain the trigonometric identities used in this solution?
3. How would the integral change if we used \( \cos^2(x) \) instead?
4. What is the geometric interpretation of the integral of \( \sin^2(x) \)?
5. How do reduction formulas work in general for different functions? 

**Tip:** Always simplify integrands using trigonometric identities before integrating for easier calculations.

Explore the integration of sin^2(x) using reduction formulas. This step-by-step solution demonstrates the use of trigonometric identities and provides the integral ∫sin^2(x)dx = (1/2)x - (1/4)sin(2x) + C. Ideal for high school calculus students.

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