An integral is a mathematical concept that represents the area under a curve in a graph. It is used to calculate the total amount or accumulation of a quantity, such as distance, velocity, or mass, over a given interval.
The process for evaluating an integral involves finding an anti-derivative of the given function, which is a function whose derivative is the original function. This can be done using various methods such as integration by substitution, integration by parts, or using a table of integrals.
This integral is used to calculate the area under the curve of a function that has the form of \frac{4}{x^2-1}. It can also be used to calculate the total amount of a quantity that follows this function, such as the total work done by a variable force.
A singularity in an integral is a point where the function being integrated becomes undefined or goes to infinity. In the integral \int\frac{4}{x^2-1}, the points x=1 and x=-1 are singularities because the function becomes undefined at these points.
The integral \int\frac{4}{x^2-1} can be evaluated using the substitution method. Let u = x^2-1, then du = 2x dx. Substituting these values into the integral gives \int\frac{4}{u}du. This can then be solved using the power rule for integration, giving the final answer of 4ln|u| + C.