Math Problem Statement

The question is in blue and the answer is in black is the answer right

Verify if the provided solution for the integral \( \int \frac{1}{\sqrt{x}} (5x - x + x^2) \, dx \) is correct.

Solution

Let me analyze the solution written and verify if the answer is correct.

The question is:

1x(5xx+x2)dx\int \frac{1}{\sqrt{x}} (5x - x + x^2) \, dx

Steps to verify:

  1. Distribute 1x\frac{1}{\sqrt{x}} across the terms inside the parentheses: 1x(5xx+x2)dx=(5xx0.5xx0.5+x2x0.5)dx\int \frac{1}{\sqrt{x}} (5x - x + x^2) \, dx = \int \left( 5x \cdot x^{-0.5} - x \cdot x^{-0.5} + x^2 \cdot x^{-0.5} \right) dx

  2. Simplify the powers of xx: 5xx0.5=5x0.5,xx0.5=x0.5,x2x0.5=x1.55x \cdot x^{-0.5} = 5x^{0.5}, \quad x \cdot x^{-0.5} = x^{0.5}, \quad x^2 \cdot x^{-0.5} = x^{1.5}

    So the integral becomes: (5x0.5x0.5+x1.5)dx\int (5x^{0.5} - x^{0.5} + x^{1.5}) \, dx

  3. Combine like terms: (4x0.5+x1.5)dx\int (4x^{0.5} + x^{1.5}) \, dx

  4. Integrate term by term:

    • For 4x0.54x^{0.5}: 4x0.5dx=4x0.5+10.5+1=4x1.51.5=83x1.5\int 4x^{0.5} \, dx = 4 \cdot \frac{x^{0.5+1}}{0.5+1} = 4 \cdot \frac{x^{1.5}}{1.5} = \frac{8}{3}x^{1.5}

    • For x1.5x^{1.5}: x1.5dx=x1.5+11.5+1=x2.52.5=25x2.5\int x^{1.5} \, dx = \frac{x^{1.5+1}}{1.5+1} = \frac{x^{2.5}}{2.5} = \frac{2}{5}x^{2.5}

  5. Combine the results: 83x1.5+25x2.5+C\frac{8}{3}x^{1.5} + \frac{2}{5}x^{2.5} + C

Now, compare with the solution written in the image:

  • The image shows partial steps and an attempt to integrate each term, but the coefficients for x1.5x^{1.5} and x2.5x^{2.5} are not properly simplified or combined.

Conclusion: The provided solution in the image is incomplete and partially correct. It does not properly simplify the coefficients, nor does it include a constant of integration CC, which is essential in indefinite integrals.

If you need clarification or further steps detailed, let me know!


Here are five related questions to enhance your understanding:

  1. Why is it important to add a constant CC in an indefinite integral?
  2. What are the common mistakes when simplifying powers of xx during integration?
  3. How can you double-check your work when integrating terms with fractional exponents?
  4. What techniques are useful for integrating terms involving x\sqrt{x}?
  5. How do you handle definite integrals differently from indefinite ones?

Tip: Always simplify your expressions completely before integrating to avoid errors in coefficients!

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Math Problem Analysis

Mathematical Concepts

Integration
Indefinite Integrals
Fractional Exponents

Formulas

Power Rule for Integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) for \(n \neq -1\)
Simplifying Exponents: \( x^a \cdot x^b = x^{a+b} \)

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12