To compute the integral of \( t \sin^2(t) \), we can use a couple of techniques. Here's the step-by-step process:

1. **Rewrite \( \sin^2(t) \)** using a trigonometric identity:
   \[
   \sin^2(t) = \frac{1 - \cos(2t)}{2}
   \]
   So, the integral becomes:
   \[
   \int t \sin^2(t) \, dt = \int t \frac{1 - \cos(2t)}{2} \, dt
   \]

2. **Separate the integral**:
   \[
   \int t \sin^2(t) \, dt = \frac{1}{2} \int t \, dt - \frac{1}{2} \int t \cos(2t) \, dt
   \]

3. **Integrate the first part**:
   \[
   \frac{1}{2} \int t \, dt = \frac{1}{2} \cdot \frac{t^2}{2} = \frac{t^2}{4}
   \]

4. **Use integration by parts** for the second integral \( \int t \cos(2t) \, dt \).

   Let:
   \[
   u = t \quad \text{and} \quad dv = \cos(2t) \, dt
   \]

   Then:
   \[
   du = dt \quad \text{and} \quad v = \frac{\sin(2t)}{2}
   \]

   Applying the integration by parts formula:
   \[
   \int u \, dv = uv - \int v \, du
   \]
   we get:
   \[
   \int t \cos(2t) \, dt = t \cdot \frac{\sin(2t)}{2} - \int \frac{\sin(2t)}{2} \, dt
   \]

   The remaining integral is:
   \[
   \int \frac{\sin(2t)}{2} \, dt = -\frac{\cos(2t)}{4}
   \]

   Therefore:
   \[
   \int t \cos(2t) \, dt = \frac{t \sin(2t)}{2} + \frac{\cos(2t)}{4}
   \]

5. **Substitute back**:
   \[
   \int t \sin^2(t) \, dt = \frac{t^2}{4} - \frac{1}{2} \left( \frac{t \sin(2t)}{2} + \frac{\cos(2t)}{4} \right)
   \]

6. **Simplify**:
   \[
   \int t \sin^2(t) \, dt = \frac{t^2}{4} - \frac{t \sin(2t)}{4} - \frac{\cos(2t)}{8} + C
   \]

Thus, the final answer is:
\[
\int t \sin^2(t) \, dt = \frac{t^2}{4} - \frac{t \sin(2t)}{4} - \frac{\cos(2t)}{8} + C
\]

Would you like more details on any step?

Here are 5 related questions to explore:

1. How do you apply integration by parts for other trigonometric integrals?
2. Can you explain the trigonometric identity for \( \cos^2(t) \)?
3. What is the general form of integration by parts?
4. How would you handle an integral involving \( \sin^3(t) \)?
5. Can we use substitution instead of parts for certain integrals?

**Tip:** When dealing with powers of trigonometric functions, using identities like the double-angle formulas simplifies the integral significantly.

Learn how to compute the integral of t * sin^2(t) using trigonometric identities and integration by parts. This step-by-step solution explains key calculus and trigonometry concepts, ideal for undergraduate students.

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