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Questions  >>  CBSE XII  >>  Math  >>  Integrals
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Q)

$\begin{align*}\int \frac{dx}{\sqrt {x^2 - a^2}} \end{align*}$


$(A)\; log|x+\sqrt{x^2+a^2}| + A$
$(B)\; log|x+\sqrt{x^2-a^2}| + A$
$(C)\; log|x+\sqrt{x+a}| + A$
$(D)\; log|x+\sqrt{x-a}| + A$

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A)
Let $x = a \sec \theta \; \; \; \implies dx = a \; \sec \theta \; \tan \theta \; d \theta$
$ \begin{align*}\int \frac{dx}{\sqrt {x^2 - a^2}} &= \int \frac {a \; \sec \theta \; \tan \theta \; d \theta}{\sqrt{a \sec^2 \theta - a^2}} \\ \int \sec \theta \; d \theta &= log | \sec \theta + \tan \theta \\ & = log \begin{vmatrix} \frac{x}{a} + \sqrt{ \frac{x^2}{a^2} + 1} \end{vmatrix} +C \\ & = log |x + \sqrt {x^2 - a^2} | - log |a| + C \\ & = log |x + \sqrt{x^2 - a^2}| +A\end{align*}$
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