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

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.