Problem 1 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The given integral is an improper integral because the integrand becomes infinite within the interval of integration [0,1].

Step 1: Check the limits of the integral

The given integral is \(\int_{0}^{1} \frac{1}{3x-2} dx\). Here both the limits, the lower limit (0) and the upper limit (1), are finite.

Step 2: Check the integrand within the limits

The function is \(\frac{1}{3x-2}\). We need to check where this function is undefined in the interval [0,1]. If the denominator, '3x-2', equals to zero then the function is undefined. We can solve '3x-2 = 0' to find when this occurs. Solving, we get the value of x as \(\frac{2}{3}\), which is between 0 and 1. So, the function indeed becomes infinite in this interval.

Step 3: Conclusion

Though both the limits of the integral are finite, the function within the limits becomes infinite at x = \(\frac{2}{3}\). Thus, the given integral is an improper integral.