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To solve the integral \(\int_{0}^{\pi} \sin^2 t \, dt\), we can use the trigonometric identity for \(\sin^2 t\):

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

Thus, the integral becomes:

\[
\int_{0}^{\pi} \sin^2 t \, dt = \int_{0}^{\pi} \frac{1 - \cos(2t)}{2} \, dt
\]

This can be split into two separate integrals:

\[
\int_{0}^{\pi} \frac{1}{2} \, dt - \int_{0}^{\pi} \frac{\cos(2t)}{2} \, dt
\]

Now, let's integrate each part separately:

1. \(\int_{0}^{\pi} \frac{1}{2} \, dt\):

\[
\int_{0}^{\pi} \frac{1}{2} \, dt = \frac{1}{2} \int_{0}^{\pi} 1 \, dt = \frac{1}{2} [t]_{0}^{\pi} = \frac{1}{2} (\pi - 0) = \frac{\pi}{2}
\]

2. \(\int_{0}^{\pi} \frac{\cos(2t)}{2} \, dt\):

\[
\int_{0}^{\pi} \frac{\cos(2t)}{2} \, dt = \frac{1}{2} \int_{0}^{\pi} \cos(2t) \, dt
\]

Let \(u = 2t\), hence \(du = 2 dt\) or \(dt = \frac{du}{2}\). The limits change as follows: when \(t = 0\), \(u = 0\); when \(t = \pi\), \(u = 2\pi\).

\[
\frac{1}{2} \int_{0}^{\pi} \cos(2t) \, dt = \frac{1}{2} \cdot \frac{1}{2} \int_{0}^{2\pi} \cos(u) \, du = \frac{1}{4} \int_{0}^{2\pi} \cos(u) \, du
\]

The integral of \(\cos(u)\) over one full period is zero:

\[
\int_{0}^{2\pi} \cos(u) \, du = [\sin(u)]_{0}^{2\pi} = \sin(2\pi) - \sin(0) = 0 - 0 = 0
\]

Therefore:

\[
\frac{1}{4} \int_{0}^{2\pi} \cos(u) \, du = \frac{1}{4} \cdot 0 = 0
\]

Putting it all together:

\[
\int_{0}^{\pi} \sin^2 t \, dt = \frac{\pi}{2} - 0 = \frac{\pi}{2}
\]

So, the value of the integral is:

\[
\int_{0}^{\pi} \sin^2 t \, dt = \frac{\pi}{2}
\]

Would you like any further details or have any questions?

Here are 5 related questions you can ask:
1. How do you integrate \(\sin^2(x)\) over a different interval?
2. What is the integral of \(\cos^2(x)\) over \([0, \pi]\)?
3. How can we use trigonometric identities to simplify integrals?
4. What is the integral of \(\sin(x)\cos(x)\) over \([0, \pi]\)?
5. How do you find the integral of a product of trigonometric functions?

**Tip:** When dealing with trigonometric integrals, using identities like the double-angle formulas can simplify the integrals significantly.

Discover how to evaluate the integral \(\int_{0}^{\pi} \sin^2 t \, dt\) using trigonometric identities and integration techniques. This detailed guide covers the use of trigonometric identities and provides a step-by-step solution, suitable for high school students.

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