# Integral Calculus - Spot the Error

• MHB
• MermaidWonders
In summary, the error is in the equation $\int_1^{\infty}\frac1x \, dx$. The comparison test for integrals requires that the integrals be on the same side of the x-axis, but your equation does not have this property.
MermaidWonders
The big blue circle has been put there by my math prof to denote the location of the error in the following solution. Why is this an error? I'm lost. :(

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$$\int_1^{\infty}\frac1x \, dx$$
does not converge. If you were to try to get an antiderivative, the only candidate is $\ln|x|$, so formally, you'd have
$$\int_1^{\infty}\frac1x \, dx = \ln|x|\big|_{1}^{\infty} =\ln(\infty)-\ln(1)=\ln(\infty)=\infty.$$
So it doesn't converge.

Ah, so is it like we're not supposed to do the "by comparison" for integrals, especially when we don't know whether the integral will be equal to infinity or not? I mean, why didn't she just circle the "converges as well" part?

Last edited:
MermaidWonders said:
Ah, so is it like we're not supposed to do the "by comparison" for integrals, especially when we don't know whether the integral will be equal to infinity or not? I mean, why didn't she just circle the "converges as well" part?

Comparison is fine, but it must be a VALID comparison. The idea would be capsulized as "closer to zero".

Bad Example: It is also true that for $$x > 1$$, we have $$-e^{x} < \dfrac{1}{x^{2}}$$, but that is not helpful.

In other words, your comparison must be on the same side of the x-axis.

... the only candidate is ln|x|...

This is a little excessive. For $$x \ge 1$$, we have $$\ln(x)$$ as an additional candidate.

Can you explain what you mean by the fact that the "comparison must be on the same side of the x-axis"?

You error is that the "comparison test" applies only to positive functions. You cannot just say that $$\displaystyle -\frac{1}{x}< \frac{1}{x^2}$$. Sometimes you can use the absolute value but it is NOT TRUE that $$\displaystyle \left|-\frac{1}{x}\right|= \frac{1}{x}< \frac{1}{x^2}$$.

Country Boy said:
You error is that the "comparison test" applies only to positive functions.

Well, BOTH positive, sure, but also BOTH negative would do.

tkhunny said:
Well, BOTH positive, sure, but also BOTH negative would do.

But "the other way around". If f(x) and g(x) are both positive with f(x)< g(x) and $\int g(x) dx$ converges then $\int f(x) dx$ converges.

If f(x) and g(x) are both negative and f(x)< g(x) (so that |g(x)|< |f(x)| and $\int f(x)dx$ converges then $\int g(x) dx$ converges.

## 1. What is integral calculus?

Integral calculus is a branch of mathematics that deals with the calculation and properties of integrals, which are mathematical objects that represent the area under a curve.

## 2. What is the difference between integral calculus and differential calculus?

The main difference between integral calculus and differential calculus is that integral calculus deals with the accumulation of quantities, while differential calculus deals with the rates at which quantities change.

## 3. How is integral calculus used in real life?

Integral calculus is used in various fields such as physics, engineering, economics, and statistics to solve problems involving continuous change and accumulation. It is used to calculate areas, volumes, and other quantities that cannot be easily calculated by basic arithmetic operations.

## 4. What are some common errors to watch out for in integral calculus?

Some common errors in integral calculus include forgetting to include the constant of integration, making algebraic mistakes, and using the wrong variable for integration. It is important to carefully check each step of the problem to avoid these errors.

## 5. How can I improve my skills in integral calculus?

To improve your skills in integral calculus, practice is key. Make sure you understand the basic concepts and techniques, and then solve a variety of problems to reinforce your understanding. You can also seek help from a tutor or online resources for additional practice and guidance.

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