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Chapter 6: Problem 1

Decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{d x}{3 x-2} $$

Short Answer

Expert verified
The integral is improper because the function \(f(x) = \frac{1}{3x-2}\) is not bounded in the interval [0,1], which makes the integral improper.

Step by step solution

01

Understanding the function

First, consider the function \(f(x) = \frac{1}{3x-2}\). This function has a singularity at \(x = \frac{2}{3}\). That means that at \(x = \frac{2}{3}\), the function goes to infinity or negative infinity, thus is not bounded.
02

Checking the limits

Now, we need to examine the limits of the integral, which are 0 and 1. Since \(\frac{2}{3}\) is within this range, the function is unbounded in the given interval
03

Conclusion

So we can conclude that the integral \(\int_{0}^{1} \frac{dx}{3x-2}\) is indeed improper because the function \(f(x) = \frac{1}{3x-2}\) is unbounded in the interval [0,1]

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