You appear to be asking for an integral involving the given terms. Let me interpret the integral as follows:

\[
\int_{\pi/4} \sin(t) \cos^2(t) \, dt
\]

To solve this:

### Step 1: Simplify the integral
Using the trigonometric identity:
\[
\cos^2(t) = 1 - \sin^2(t)
\]
The integral becomes:
\[
\int_{\pi/4} \sin(t) (1 - \sin^2(t)) \, dt
\]

### Step 2: Use substitution
Let:
\[
u = \sin(t) \quad \text{so that} \quad du = \cos(t) \, dt
\]

Change the limits of integration for \( u \):
- When \( t = \pi/4 \), \( u = \sin(\pi/4) = \frac{\sqrt{2}}{2} \).

Thus, the integral becomes:
\[
\int \sin(t) (1 - \sin^2(t)) \, dt = \int_{\sqrt{2}/2}^{u} u(1 - u^2) \, du
\]

### Step 3: Expand and integrate
Expand \( u(1 - u^2) \):
\[
\int_{\sqrt{2}/2} u(1 - u^2) \, du = \int_{\sqrt{2}/2} (u - u^3) \, du
\]

Now integrate each term:
\[
\int u \, du = \frac{u^2}{2}, \quad \int u^3 \, du = \frac{u^4}{4}
\]

Thus:
\[
\int_{\sqrt{2}/2} u(1 - u^2) \, du = \left[\frac{u^2}{2} - \frac{u^4}{4} \right]_{\sqrt{2}/2}^1
\]

### Step 4: Evaluate the limits
First, calculate at \( u = 1 \):
\[
\frac{1^2}{2} - \frac{1^4}{4} = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}
\]

Now, calculate at \( u = \frac{\sqrt{2}}{2} \):
\[
\frac{\left(\frac{\sqrt{2}}{2}\right)^2}{2} - \frac{\left(\frac{\sqrt{2}}{2}\right)^4}{4} = \frac{\frac{2}{4}}{2} - \frac{\frac{4}{16}}{4} = \frac{1}{4} - \frac{1}{16} = \frac{4}{16} - \frac{1}{16} = \frac{3}{16}
\]

Subtract the results:
\[
\frac{1}{4} - \frac{3}{16} = \frac{4}{16} - \frac{3}{16} = \frac{1}{16}
\]

### Final Answer:
\[
\int_{\pi/4} \sin(t) \cos^2(t) \, dt = \frac{1}{16}
\]

---

Do you need further details or clarifications?

### Expanding Questions:
1. Would you like to verify each substitution step?
2. Do you want a graph illustrating this integral geometrically?
3. Shall I explain the trigonometric identity \( \cos^2(t) = 1 - \sin^2(t) \) in detail?
4. Would you like to explore alternative methods to solve this integral?
5. Are you interested in practical applications of such integrals?

### Tip:
Always double-check the limits of integration after substitution to avoid errors in definite integrals!

Learn how to solve the definite integral of sin(t)cos²(t) from π/4 with a detailed, step-by-step solution. This problem involves integration by substitution and uses trigonometric identities. Suitable for advanced high school students studying calculus.

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