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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate : $\int \sqrt {e^x-1} dx$

$(a)\;2 \sqrt {e^x-1}-2 \tan ^{-1}(a \sqrt {e^{x}-1})+c\qquad(b)\;2 \sqrt {e^x-1}+c \qquad(c)\;-2 \cos ^{-1} \sqrt {e^x-1}+c \qquad (d)\;None$

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1 Answer

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$\sqrt {e^x-1}=t$
$\qquad= \large\frac{1}{2 \sqrt {e^x-1}}$$ \times e^x .dx=dt$
=> $dx= \large\frac{2 \sqrt {e^x-1}}{e^x}$$.dt$
=> $ dx= 2. \large\frac{t}{(t^2+1)}$$dt$
$dx= \large\frac{2t}{(1+t)^2}$$ dt$
=> $\int \large\frac{2t^2}{ [(1) +(t)^2 ]}$$dt$
=> $2 \int \large\frac{t^2 +1}{t^2+1} -\frac{1}{1+t^2}$$dt$
=> $ 2t -2 \tan ^{-1} (t) +c$
=>$2 \sqrt {e^x-1}-2 \tan ^{-1}(a \sqrt {e^{x}-1})+c$
Hence a is the correct answer.
answered Dec 24, 2013 by meena.p
 
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