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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate : $\int \sqrt {(\sec x -1)}.dx$

$(a)\;\sec x -1+c \qquad(b)\; \cos x -1+c \qquad(c)\;-2 \log | \sqrt {2}\cos \frac{x}{2}+ \cos x |+c\qquad (d)\;None$

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1 Answer

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$\int \sqrt {(\sec x -1)}.dx$
=> $\int \large\frac{\sqrt {1- \cos x }}{\sqrt {\cos x }}$$.dx$
=> $\int \large\frac {\sqrt 2 \sin (x/2)dx}{\sqrt {2 \cos ^2 (x/2) -1}}$
Suppose =>$ \sqrt 2 \cos (x/2) =t$
$\large\frac{-1}{2}$$ \times \sqrt 2 \sin (x/2) dx=dt$
=> $\large\frac{-1}{2} $$ \sin (x/2) dx=dt$
Putting this values we get,
=> $ \int \large\frac{(-2) dt}{\sqrt {t^2-1}}$
=>$-2 \int \large\frac{1}{\sqrt {t^2 -1}}$$dt$
=> $ -2 \log |(t+\sqrt {t^2-1)}|+c$
=> $ -2 \log |(\sqrt 2 \cos \frac{x}{2} +\sqrt {2 \cos ^2 \frac{x}{2} -1})|+c$
=> $ -2 \log |\sqrt 2 \cos \frac{x}{2} + \cos x |+c$
Hence c is the correct answer.
answered Dec 23, 2013 by meena.p
 
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