Integral of 1 over x squared plus 1 Using Partial Fractions

Evaluate the integral of $\displaystyle \frac{1}{x^2 + 1}$ using partial fractions.

When tackling the integral of a rational function, one of the effective approaches is applying the method of partial fraction decomposition. This method is particularly useful when you are dealing with complex rational expressions that can be decomposed into simpler fractions. In this problem, however, the function $\frac{1}{x^2 + 1}$ is already in a form that doesn't lend itself directly to partial fraction decomposition in the traditional sense, as it is not a proper rational function where the degree of the numerator is less than the degree of the denominator. In such cases, it is important to recognize the structure of the denominator and identify if other integration techniques might be more appropriate.

In the context of this function, noticing the denominator $x^2 + 1$ might suggest a trigonometric substitution approach, specifically recognizing it as a trigonometric identity related to the arctan function. The derivative of arctan $(\arctan x)$ is exactly $\frac{1}{x^2 + 1}$, indicating that the integral can directly be evaluated to be the arctangent function. Thus, recognizing this functional form can simplify the integration process significantly, bypassing the need for partial fraction decomposition.

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