The correct option is D 1/b^ 1(ax/b)+c ∫1x√(a^2x^2 b^2)dx =∫dxbx√(a^2x^2b^2 1) =1b∫1· #160; dx(ba)(axb)√(a^2x^2b^2 1)=1b^ 1(axb)+c. Hence ...

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Integration of Irrational Algebraic Fractions - 2

∫1/x√a2x2-b2d...

Question

$\int \frac{1}{x \sqrt{a^{2} x^{2} - b^{2}}} d x =$

\frac{1}{b} {sec}^{- 1} x + c

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\frac{1}{b} {sec}^{- 1} (a x) + c

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\frac{1}{b} {sec}^{- 1} (\frac{a x}{b}) + c

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\frac{- 1}{b} {sec}^{- 1} (\frac{a x}{b}) + c

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Solution

The correct option is D $\frac{1}{b} {sec}^{- 1} (\frac{a x}{b}) + c$
$\int \frac{1}{x \sqrt{a^{2} x^{2} - b^{2}}} d x$
$= \int \frac{d x}{b x \sqrt{\frac{a^{2} x^{2}}{b^{2}} - 1}}$
$= \frac{1}{b} \int \frac{1 \cdot d x}{(\frac{b}{a}) (\frac{a x}{b}) \sqrt{\frac{a^{2} x^{2}}{b^{2}} - 1}} = \frac{1}{b} {sec}^{- 1} (\frac{a x}{b}) + c .$
Hence, option 'C' is correct.

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