There is a method. They are completing the square in the quadratic in the denominator.TheKShaugh said:Homework Statement
[tex]\displaystyle{\int}\dfrac{x}{\sqrt{x^2+x+1}}~dx =
\displaystyle{\int}\dfrac{(x+\frac 1 2)-\frac 1 2}{\sqrt{(x+\frac 1 2)^2+(\frac {\sqrt 3} 2)^2}}~dx[/tex]
Homework Equations
Given.
The Attempt at a Solution
The solution is given, but I'm not sure how it was found. Is there a method for finding those factors or is it trial and error/intuition?
"U sub" is a common abbreviation for the method of substitution in calculus. It involves choosing a new variable, u, to substitute in for a more complicated expression. This is often used when finding factors because it can simplify the expression and make it easier to factor.
"U sub" is typically used when the expression being factored involves a polynomial or a complicated function. This method can help to simplify the expression and make it easier to factor by breaking it down into smaller parts.
While "U sub" can be a helpful tool in finding factors, it is not always necessary or applicable. It is most commonly used for polynomial or complicated expressions, but may not be useful for simpler expressions. It is important to assess the expression and determine if "U sub" would be helpful in finding factors.
One limitation of "U sub" is that it may not always work for every expression. Some expressions may not be able to be simplified using this method, or it may lead to a more complicated expression. It is important to consider other factoring methods and choose the most appropriate one for each individual expression.
The method of "U sub" makes finding factors easier by simplifying the expression and breaking it down into smaller parts. By substituting a more complicated expression with a simpler one, it can make the factoring process more manageable. This method can also be helpful in identifying common factors and patterns within the expression.