Your first thought led you to try an ordinary substitution, u = x^2 + 1. This won't work, though, because du = 2xdx, so there's no way to change the given integrand to du/u.thereddevils said:Homework Statement
evaluate [tex]\int \frac{1}{x^2+1} dx[/tex]
Homework Equations
The Attempt at a Solution
This can't be [tex]\frac{\ln x^2+1}{2x}[/tex] , my first thought on this .
Then , i tried partial fraction , it didn't work either .
The integral of 1/(x^2+1) can be solved by using the substitution method. Let u = x^2+1, then du = 2x dx. The integral becomes ∫ 1/u * (1/2) du which is equal to (1/2) * ln|u| + C. Substituting back x^2+1 for u, the final answer is (1/2) * ln|x^2+1| + C.
The domain of 1/(x^2+1) includes all real numbers except for x = ±i, where i is the imaginary unit. This is because when x = ±i, the denominator becomes 0, which is undefined.
No, the power rule can only be used for functions in the form of x^n, where n is a real number. 1/(x^2+1) cannot be written in this form, so the power rule cannot be used to solve its integral.
No, there is no shortcut or trick to solve the integral of 1/(x^2+1). The most efficient way to solve it is by using the substitution method as described in the first question.
The integral of 1/(x^2+1) can be used in various fields such as physics, engineering, and economics. For example, it can be used to calculate the area under a curve in statistics and to find the center of mass in physics. In economics, it can be used to calculate the utility function of a consumer.