Integration is a mathematical process that involves finding the area under a curve in a graph. It is the inverse operation of differentiation and is used to solve problems related to finding the total value of a function over a given interval.
This integration problem is a common example used to demonstrate the process of integration. It helps to understand the basic concepts and techniques of integration, which are essential for solving more complex problems in mathematics and science.
The first step is to factor out a 1/2 from the denominator to get 1/2(1/x+x). Then, use the substitution method by letting u=1/x. After substituting and simplifying, the integral becomes 1/2∫(1/u+u)du. This can be solved by using the natural log and power rules of integration.
The limits of integration depend on the specific problem or context in which the integration is being used. If there are no given limits, the integral is considered to be indefinite and will result in a general solution with a constant of integration.
You can check your answer by differentiating the result and seeing if it simplifies back to the original function 1/(2x+2x^2). You can also use online integration calculators to verify your solution or ask a math tutor or professor for help.