To evaluate the integral [tex]\(\int \frac{1}{x^2 + 1} \, dx\)[/tex], we can proceed by recognizing that this integral has a standard form. Let's go through the steps in detail:
1. Recognize the standard form:
The given integral [tex]\(\int \frac{1}{x^2 + 1} \, dx\)[/tex] is of the form [tex]\(\int \frac{1}{a^2 + x^2} \, dx\)[/tex], where [tex]\(a = 1\)[/tex].
2. Identify the antiderivative:
The integral [tex]\(\int \frac{1}{a^2 + x^2} \, dx\)[/tex] is a standard integral that results in [tex]\(\frac{1}{a} \arctan \left(\frac{x}{a}\right) + C\)[/tex]. When [tex]\(a = 1\)[/tex], this simplifies to:
[tex]\[
\int \frac{1}{x^2 + 1} \, dx = \arctan(x) + C
\][/tex]
3. Answer:
Therefore, the evaluated integral is:
[tex]\[
\int \frac{1}{x^2 + 1} \, dx = \arctan(x) + C
\][/tex]
Here, [tex]\(C\)[/tex] is the constant of integration.
