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To solve the integral 

\[
\int \left(3t^2 + \frac{t}{2}\right) dt,
\]

we can integrate each term separately.

### Step 1: Integrate \(3t^2\)
The integral of \(3t^2\) with respect to \(t\) is:
\[
\int 3t^2 \, dt = 3 \cdot \frac{t^3}{3} = t^3.
\]

### Step 2: Integrate \(\frac{t}{2}\)
The integral of \(\frac{t}{2}\) with respect to \(t\) is:
\[
\int \frac{t}{2} \, dt = \frac{1}{2} \cdot \frac{t^2}{2} = \frac{t^2}{4}.
\]

### Step 3: Combine the results
Adding both parts, we get:
\[
\int \left(3t^2 + \frac{t}{2}\right) dt = t^3 + \frac{t^2}{4} + C,
\]
where \(C\) is the constant of integration.

Would you like a more detailed explanation on any part of this solution?

### Related Questions:
1. How do you integrate a polynomial function term-by-term?
2. What are some techniques for integrating expressions with fractions?
3. How is the constant of integration important in indefinite integrals?
4. Can we verify this integral result by differentiating the answer?
5. How does the integral change if we have limits on the integral sign?

**Tip:** When integrating, handling each term separately can simplify the process, especially with polynomial expressions.

Learn how to find the integral of the expression 3t^2 + t/2 step-by-step. This solution covers polynomial integration techniques and uses the power rule for integration. Suitable for high school students in Grades 11-12.

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