∫ ^3 _0 (t^2 + 2)dt Rewrite it so that it following the steps ∫ ^3 _0 (1 + 1/2 t^2)^2 dt = [t + ____________] ^3 _0

To rewrite the integral in the form given, we will first expand and simplify. Let's start from the original integral:

\[
\int_{0}^{3} \left(t^2 + 2\right) \, dt
\]

The goal is to rewrite this in a way that matches the form:

\[
\int_{0}^{3} \left(1 + \frac{1}{2}t^2\right)^2 \, dt
\]

1. **Expand \(\left(1 + \frac{1}{2}t^2\right)^2\):**

   \[
   \left(1 + \frac{1}{2}t^2\right)^2 = 1 + 2 \cdot \frac{1}{2}t^2 + \left(\frac{1}{2}t^2\right)^2
   \]

2. **Simplify:**

   \[
   = 1 + t^2 + \frac{1}{4}t^4
   \]

Thus, the integral now becomes:

\[
\int_{0}^{3} \left(1 + t^2 + \frac{1}{4}t^4\right) \, dt
\]

3. **Integrate term by term:**

   \[
   \int_{0}^{3} \left(1 + t^2 + \frac{1}{4}t^4\right) \, dt = \left[t + \frac{t^3}{3} + \frac{t^5}{20}\right]_{0}^{3}
   \]

Would you like to proceed with calculating this expression, or do you need more details on any specific step?

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1. How do we expand expressions with exponents?
2. How does each term in the expansion affect the integral?
3. What is the process for evaluating definite integrals?
4. How can we simplify complex expressions before integrating?
5. What are the practical applications of integrating polynomials?

**Tip:** When expanding, use binomial formulas to quickly rewrite expressions raised to powers.

∫ ^3 _0 (t^2 + 2)dt Rewrite it so that it follows the steps ∫ ^3 _0 (1 + 1/2 t^2)^2 dt = [t + ____________] ^3 _0

Discover how to rewrite and solve the integral of (t^2 + 2) over [0, 3] in terms of a transformed expression (1 + 1/2 t^2)^2. This problem provides step-by-step guidance on polynomial expansion, integral setup, and term-by-term integration, making it ideal for students studying integral calculus.

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