…m1ke_ said:
The general formula for integrating arctan(y) dx is ∫arctan(y) dx = y*arctan(y) - ln|y| + C, where C is the constant of integration.
To solve the integral of arctan(y) dx, you can use integration by parts or substitution. The latter involves substituting u = arctan(y) and du = dy/(1+y^2), which simplifies the integral to ∫u du. From there, you can use the power rule to integrate and then substitute back in for u to get the final answer.
Yes, arctan(y) dx can be integrated using trigonometric identities, specifically the identity tan(arctan(y)) = y and the inverse derivative formula d/dx(arctan(x)) = 1/(1+x^2). These identities can be used in conjunction with substitution to simplify the integral and solve for the final answer.
The domain of integration for arctan(y) dx is all real numbers except for y = -1 and y = 1, since these values would result in division by zero in the integral's solution. Additionally, the domain should also not include any points where y^2 + 1 = 0, as this would also result in division by zero.
Yes, the integral of arctan(y) dx can be evaluated numerically using methods such as the trapezoidal rule or Simpson's rule. These methods involve approximating the integral using a series of smaller subintervals and then summing them to get an approximate value for the integral. This is useful for cases where the integral cannot be solved analytically.