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Properties Derived from Trigonometric Identities

∫sin -12 × 1+...

Question

$\int \sin^{- 1} (\frac{2 x}{1 + x^{2}}) d x$

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Solution

$\int \sin^{- 1} (\frac{2 x}{1 + x^{2}}) d x Let x = \tan θ d x = \sec^{2} θ d θ ∴ \int \sin^{- 1} (\frac{2 x}{1 + x^{2}}) d x = \int \sin^{- 1} (\frac{2 \tan θ}{1 + \tan^{2} θ}) . \sec^{2} θ d θ = \int \sin^{- 1} (\sin 2 θ) . \sec^{2} θ d θ = \int (2 θ) \sec^{2} θ d θ = 2 \int \underset{I}{θ} \underset{II}{\sec^{2} θ} d θ = 2 [θ \int \sec^{2} θ d θ - \int \{\frac{d}{d θ} (θ) \int s {ec}^{2} θ d θ\} d θ] = 2 [θ . \tan θ - \int 1 . \tan θ d θ] = 2 [θ \tan θ - \log |\sec θ|] + C = 2 [θ \tan θ - \log {|1 + \tan^{2} θ|}^{\frac{1}{2}}] + C = 2 [(\tan^{- 1} x) \times x - \log {(1 + x^{2})}^{\frac{1}{2}}] + C = 2 x \tan^{- 1} x - 2 \times \frac{1}{2} \log |1 + x^{2}| + C = 2 x \tan^{- 1} x - \log |1 + x^{2}| + C$

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