First slide

Introduction to integration

Question

$\int \cos^{- 1} (\frac{1}{x}) dx$ is equal to

Moderate

A

${xsec}^{- 1} x + \cosh^{- 1} x + C$
B

${xsec}^{- 1} x - \cosh^{- 1} x + C$
C

${xsec}^{- 1} x - \sin^{- 1} x + C$
D

None of the above

Solution

$\int \cos^{- 1} (\frac{1}{x}) dx = \int \sec^{- 1} x \cdot 1 dx$
$\begin{array}{l} = \sec^{- 1} x \int dx - \int [\frac{d}{dx} \sec^{- 1} x \int dx] dx \\ = {xsec}^{- 1} x - \int \frac{1}{x \sqrt{x^{2} - 1}} xdx \\ = {xsec}^{- 1} x - \int \frac{1}{\sqrt{x^{2} - 1}} dx \\ = {xsec}^{- 1} x - \cosh^{- 1} x + C \end{array}$

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