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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate :$\int \limits_0^1 \large\frac{1}{\sqrt {x-(z-x)}} $$dx$

\[\begin {array} {1 1} (a)\;z \\ (b)\;\frac{z}{2} \\ (c)\;2z \\ (d)\;z^2 \end {array}\]
Can you answer this question?
 
 

1 Answer

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$\int \limits_{\alpha}^{\beta} \large\frac{1}{\sqrt {(\alpha-\beta)(\beta-\alpha)}} $$dx=\pi, \beta > \alpha$
$\alpha=0$
$\beta=1$
By solving this
$\qquad= z$
another method is
=> $\int \limits_0^1 \large\frac{1}{\sqrt {x(1-x)}}$$dx$
=> $\int \limits_0^1 \large\frac{1}{x-x^2+\Large\frac{1}{4}-\frac{1}{4}}$$dx$
$\int \limits_0^1 \large\frac{1}{\sqrt {(\Large\frac{1}{2})^2-(x - \Large\frac{1}{2})^2}} $$dx$
$\qquad = \bigg[\sin ^{-1} \bigg( \large\frac{x-1/2}{1/2}\bigg)\bigg]_0^1$
$\qquad= \sin ^{-1} (1) +\sin ^{-1}(1)$
$\qquad=z$
answered Dec 12, 2013 by meena.p
 
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