What is the result of the integral ∫ dx/(x^2 + 1)?

The integral of ( \frac{1}{x^2 + 1} ) is a standard result in calculus that leads to the arctangent function. Specifically, the integral can be evaluated using the substitution ( x = \tan(t) ), leading to the result that:

[

\int \frac{1}{x^2 + 1} , dx = \arctan(x) + C

]

Here, the arctangent function ( \arctan(x) ) is defined as the inverse of the tangent function, giving the angle whose tangent is ( x ). This connection is crucial because the derivative of the arctangent function is exactly ( \frac{1}{x^2 + 1} ), thereby making it the appropriate antidote for the integral.

The other functions mentioned in the choices—arccos, natural logarithm, and sine—do not relate to the specific form of the function given in the integral and thus do not yield ( \frac{1}{x^2 + 1} ) when differentiated. Consequently, the integration of the rational function ( \frac{1}{x^2 + 1} ) clearly points to the ar