Please solve the follwoing intergal determine whether the integral converges or diverges. If the integral converges, evaluate it.
Transcribed Image Text:The image shows an improper integral represented as follows:
\[
\int_{1}^{\infty} \frac{\ln x}{x^2} \, dx
\]
This integral represents the area under the curve \( y = \frac{\ln x}{x^2} \) from \( x = 1 \) to \( x = \infty \). The integrand function \(\frac{\ln x}{x^2}\) involves the natural logarithm of \( x \) divided by \( x \) squared.
Key points to consider about this integral:
1. **Improper Integral**: This integral is improper because it has an infinite upper limit of integration.
2. **Behavior of the Integrand**: As \( x \to \infty \), the function \(\frac{\ln x}{x^2}\) approaches zero, which influences the convergence of the integral.
3. **Convergence**: Determining whether this integral converges requires analysis, such as using an appropriate convergence test for improper integrals.
Understanding and evaluating this integral involves techniques related to improper integrals, integration by parts, and convergence tests.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
![The image shows an improper integral represented as follows:
\[
\int_{1}^{\infty} \frac{\ln x}{x^2} \, dx
\]
This integral represents the area under the curve \( y = \frac{\ln x}{x^2} \) from \( x = 1 \) to \( x = \infty \). The integrand function \(\frac{\ln x}{x^2}\) involves the natural logarithm of \( x \) divided by \( x \) squared.
Key points to consider about this integral:
1. **Improper Integral**: This integral is improper because it has an infinite upper limit of integration.
2. **Behavior of the Integrand**: As \( x \to \infty \), the function \(\frac{\ln x}{x^2}\) approaches zero, which influences the convergence of the integral.
3. **Convergence**: Determining whether this integral converges requires analysis, such as using an appropriate convergence test for improper integrals.
Understanding and evaluating this integral involves techniques related to improper integrals, integration by parts, and convergence tests.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1065c73-a773-4439-b194-4cac2295760b%2F5f7570d2-6478-4511-9f05-43b545a514f8%2Fkgm0yo6_processed.png&w=3840&q=75)
