https://r2.erweima.ai/mathbot/image/1725594997353xsk67j4e.jpg

The image contains an assignment titled "Asignación 1 Ing Sistema," with an integral to solve in part 1. Specifically, the problem asks to compute the integral:

\[
\int \sqrt{t} \, dt
\]

It also includes steps for "Reescribir" (rewrite), "Integrar" (integrate), and "Simplificar" (simplify). Let’s go through these steps:

1. **Reescribir (Rewrite):**
   The integral of \(\sqrt{t}\) can be rewritten using exponent notation:
   \[
   \int t^{1/2} \, dt
   \]

2. **Integrar (Integrate):**
   To integrate \(t^{1/2}\), apply the power rule for integration, which states:
   \[
   \int t^n \, dt = \frac{t^{n+1}}{n+1} + C
   \]
   Here, \(n = 1/2\), so:
   \[
   \int t^{1/2} \, dt = \frac{t^{(1/2) + 1}}{(1/2) + 1} + C = \frac{t^{3/2}}{3/2} + C = \frac{2}{3} t^{3/2} + C
   \]

3. **Simplificar (Simplify):**
   The simplified result is:
   \[
   \frac{2}{3} t^{3/2} + C
   \]

If you'd like more details or have any questions, let me know! Here are five related questions to expand on this topic:

1. How does the power rule for integration work in general?
2. What happens when you integrate negative exponents?
3. Can you apply substitution methods to integrals with roots?
4. What is the relationship between differentiation and integration?
5. How do you integrate functions with fractional exponents?

**Tip:** Always double-check the exponent before applying the power rule to avoid calculation mistakes.

Dado los siguientes ejercicios completar la tabla. Integral: ∫√t dt.

Learn how to solve the integral ∫√t dt by rewriting it using exponent notation and applying the power rule for integration. This step-by-step guide explains the process of rewriting, integrating, and simplifying the expression, suitable for calculus students.

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