First slide

Introduction to integration

Question

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

Moderate

A

$x \tan^{- 1} x - \ln |\sec (\tan^{- 1} x)| + C$
B

$x \tan^{- 1} x + \ln |\sec (\tan^{- 1} x)| + C$
C

$x \tan^{- 1} x - \ln |\cos (\tan^{- 1} x)| + C$
D

$None of these$

Solution

$\begin{array}{l} I = \int \sin^{- 1} (\frac{2 x}{1 + x^{2}}) dx, let x = \tan θ \\ \Rightarrow dx = \sec^{2} θdθ \\ ∴ I = \int \sin^{- 1} (\frac{2 \tan θ}{1 + \tan^{2} θ}) \cdot \sec^{2} θdθ = 2 \int θ \cdot \sec^{2} θdθ \\ = 2 (θ \cdot \tan θ - \ln ∣ \sec θ) + C \\ = 2 (x \cdot \tan^{- 1} x - \ln |\sec (\tan^{- 1} x)|) + C \end{array}$

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