dx-1-x-2-

Question Number 92138 by hore last updated on 05/May/20

\int \frac{d x}{1 + x^{2}}

Answered by john santu last updated on 05/May/20

\tan^{- 1} (x) + c

Answered by Rio Michael last updated on 07/May/20

\int \frac{d x}{1 + x^{2}}

let x = \tan θ \Rightarrow \frac{d x}{d θ} = \sec^{2} θ

\Rightarrow \int \frac{d x}{1 + x^{2}} = \int \frac{1}{1 + \tan^{2} θ} \sec^{2} θ d θ

= \int \frac{\sec^{2} θ}{\sec^{2} θ} d θ_{1 + \tan^{2} θ = \sec^{2} θ}

= \int d θ = θ + C

but x = \tan θ \Rightarrow θ = \tan^{- 1} x

\Rightarrow \int \frac{d x}{1 + x^{2}} = \tan^{- 1} x + C