Understand Comparison Theorem For Improper Integral Ex3 | Summary and Q&A

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March 30, 2015
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blackpenredpen
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Understand Comparison Theorem For Improper Integral Ex3

TL;DR

The improper integral from 1 to infinity of x over square root of x plus 5 plus 1 dx converges.

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Key Insights

  • πŸ“ The comparison rule is a useful tool in determining if an improper integral converges or diverges.
  • πŸ˜‘ Simplifying the expression using exponents helps in analyzing the behavior of the integral as x approaches infinity.
  • ☺️ The power value associated with the x term plays a crucial role in determining the convergence of the integral.
  • 😫 Setting up inequalities and comparing the given integral with known convergent or divergent integrals helps draw conclusions.
  • βœ… Following a structured approach of selecting a known convergent integral, checking the inequality, and drawing a conclusion is essential in determining convergence.
  • ☺️ The highest powers of x on the top and bottom can be compared to simplify and analyze the behavior of the integral.
  • ✊ The power value of three half, which is greater than one, indicates convergence in the given integral.

Transcript

so this is the question i would like to answer does the in-property integral the integral from 1 to infinity x over square root of x plus 5 plus 1 instead of the square root dx does that converge or not here is the idea i cannot really integrate this from scratch because i think that would be really helpful maybe it's not possible but then we would... Read More

Questions & Answers

Q: How can we determine if an improper integral converges or not?

To determine if an improper integral converges, we can use the comparison rule. If we can show that the given integral is less than or equal to a known convergent integral, it will also converge.

Q: How does the content simplify the given improper integral?

The content simplifies the improper integral by analyzing the highest powers of x on the top and bottom. By subtracting the exponents, it is shown that the given expression is approximately equal to 1 over x plus 3/2 as x approaches infinity.

Q: What condition should be met for an improper integral to converge?

For an improper integral to converge, the power value associated with the x term should be greater than 1. In this case, the power is three half, which is greater than one, indicating convergence.

Q: How does the content determine the convergence of the improper integral using the comparison theorem?

The content compares the given integral to a known convergent integral by setting up an inequality. By cross multiplying and simplifying, it is shown that the given integral is less than or equal to the converging integral. Hence, it can be concluded that the improper integral converges.

Summary & Key Takeaways

  • The content discusses how to determine whether the improper integral converges or not using the comparison rule.

  • If the improper integral is less than or equal to a known convergent integral, it will also converge, and if it is greater than or equal to a known divergent integral, it will diverge.

  • The content simplifies the given improper integral and compares it to a known converging integral to conclude that it converges.

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