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

The integral of ( \frac{1}{1+x^2} , dx ) is a standard result in calculus, particularly in integral calculus involving trigonometric and inverse trigonometric functions.

To understand why the correct answer is ( \text{arctan}(x) + C ), consider the fact that the function ( \frac{1}{1+x^2} ) is the derivative of ( \text{arctan}(x) ). When you differentiate ( \text{arctan}(x) ), you apply the chain rule, yielding:

[

\frac{d}{dx}(\text{arctan}(x)) = \frac{1}{1+x^2}

]

Thus, integrating ( \frac{1}{1+x^2} ) naturally leads to the ( \text{arctan}(x) ) function. The constant ( C ) represents the constant of integration, which is essential in indefinite integrals to represent the family of antiderivatives.

Other options do not yield the correct result upon integration. For instance, ( \text{tan}(x) ) is related to the function ( \frac{d}{dx