Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Use Improper
![**Example Problem 1:**
Evaluate the improper integral:
\[ \int_{e^2}^{\infty} \frac{dx}{x \ln^4(x)} \]
---
**Explanation:**
This integral involves the evaluation of an improper integral with bounds from \(e^2\) to infinity. The function to be integrated is \(\frac{1}{x \ln^4(x)}\). The solution of this integral may involve techniques such as substitution and understanding its convergence or divergence based on the behavior of the integrand as \(x\) approaches infinity.
---
**Graph/Diagram:**
There are no graphs or diagrams for this problem, but typically, one might consider plotting the function \( \frac{1}{x \ln^4(x)} \) to understand its behavior. For this specific problem, as \(x \to \infty\), the integrand \(\frac{1}{x \ln^4(x)}\) tends to zero, and as \(x \to e^2\), the integrand has finite value since the logarithm of a positive value (in this case, \( e^2 \)) is defined.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a36eedc-1311-4ac7-908a-8a22bc0a900c%2F392827f9-dfca-4762-9778-7d20c9f1c2c9%2Frde0hm_processed.png&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
