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Find the Integral 1/(x^2+1)

In the realm of calculus, certain integrals stand out for their frequent application and the elegant techniques required for their resolution. One such integral is 1/(x^2+1), notable for its direct connection to the arctangent function, a fundamental element in trigonometric calculations. The process of integrating 1/(x^2+1) showcases the synergy between algebraic expressions and trigonometric identities, simplifying complex calculations elegantly.

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

Solution:

Let\ x=tan(\theta)⇒dx=\sec^2(\theta) d\theta \& \theta=tan^{−1}(x)

\therefore \int \frac{1}{x^2+1} \, dx = \int \frac{1}{\tan^2(\theta) + 1} \sec^2(\theta) \, d\theta

= \int \sec^2(\theta) \, d\theta

= \int d\theta

= \theta + C

= \tan^{-1}(x) + C