https://r2.erweima.ai/mathbot/image/1725585272078804k4cjt.jpg

The problem asks to complete a table with different steps for the integral \( \int \sqrt{x} \, dx \). Here’s a detailed breakdown for each step:

1. **Rewrite**:
   Convert the square root function into an exponent form:
   \[
   \sqrt{x} = x^{1/2}
   \]

2. **Integrate**:
   Apply the power rule for integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n \neq -1 \).
   \[
   \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3} x^{3/2} + C
   \]

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

Thus, the completed table would have the following entries:

- **Original Integral**: \( \int \sqrt{x} \, dx \)
- **Rewrite**: \( \int x^{1/2} \, dx \)
- **Integrate**: \( \frac{2}{3} x^{3/2} + C \)
- **Simplify**: \( \frac{2}{3} x^{3/2} + C \)

Would you like further explanation or have any questions?

Here are some related questions to expand your understanding:

1. How does the power rule for integration work with negative exponents?
2. What is the integral of \( \int x^{-1/2} \, dx \)?
3. How do we handle integrals with fractional exponents in general?
4. What is the connection between integration and differentiation for polynomial terms?
5. How would the process differ if we integrated \( \int x^{1/2} \, dx \) over a specific interval?

**Tip**: When faced with integrals of roots or fractional powers, always rewrite them in exponential form for easier integration.

Given the following exercises, complete the table for the integral ∫√x dx.

Learn how to integrate the square root of x (√x) with a detailed step-by-step explanation. This problem involves key calculus concepts such as the power rule for integration and exponent manipulation, suitable for advanced high school or college-level calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation