Problem 19 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{1}^{\infty} \ln x d x\)

Step-by-Step Solution

Answer

The integral \( \int_{1}^{\infty} \ln x dx \) is divergent.

Step 1: Find the Antiderivative

To evaluate this integral, first consider the antiderivative of \( \ln x \) which turns out to be \( x \ln x - x \).

Step 2: Evaluate the Antiderivative at Boundaries

The next step would be to evaluate this antiderivative at the boundaries of integration (1 and \( \infty \)). To do this, compute \( \lim _{x \rightarrow \infty} (x \ln x - x) - (1 \ln 1 - 1) \).

Step 3: Calculate Limit as x Approaches Infinity

Compute the limit mentioned in the previous step. If the limit exists, i.e., it approaches a finite value, the integral is convergent. If the limit is infinity or doesn't exist, the integral is divergent. In this case, one can apply l'Hopital's rule to calculate \( \lim _{x \rightarrow \infty} (x \ln x - x) \).

Step 4: Final Conclusion

By calculating the limit, we find that the value is negative infinity. Hence, we can conclude that the integral \( \int_{1}^{\infty} \ln x dx \) is divergent.