What Formula Do You Use for the Integral of Inverse Tangent?

The formula for the integral of inverse tangent is the integral of arctan(x) dx = x * arctan(x) - (1/2) * ln |x2+1|+ C. The integral is solved using integration by parts, which notes that the integral of u dv is equal to u times v minus the integral of v du. The term arctan represents the inverse function in mathematical formulas.

Using integration by parts for the integral of inverse tangent, the variable u is set to arctan(x), which means that the derivative of u, expressed as du, is set to 1/((x^2)+1). The derivative dv is set to dx, which when integrated, sets v equal to x. The integral of arctan(x) is rewritten as x * arctan(x) minus the integral of x/((x^2) + 1) dx. Integrating v du gives u = (x^2) + 1. The derivative of u, du, is 2xdx, from which the formula x dx = du/2 can be derived.

From those values, the integral of (x/(x^2) + 1) dx is evaluated as 1/2u du = (1/2) ln |u| = (1/2) ln |(x^2) + 1|. In these formulas, the line symbols indicate an absolute value that is neither positive nor negative. For the final solution, x * arctan(x) - (1/2) * ln |x2+1|+ C, C represents the constant of integration.