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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate: $\int \limits_0^{\rho} \large\frac{y.dy}{\sqrt {y+\rho}}$

\[\begin {array} {1 1} (a)\; \frac{2}{3} (2 - \sqrt 2) \lambda \sqrt {\lambda} \\ (b)\;(2- \sqrt 2) \lambda \sqrt {\lambda} \\ (c)\;\lambda \sqrt {\lambda} \\ (d)\;None \end {array}\]

Can you answer this question?
 
 

1 Answer

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$\int \limits_0^{\rho} \sqrt { y + \lambda} dy- \lambda \sqrt {\lambda}$
=> $y+ \rho=t^2$
=> $ dy =2t dt$
$\int\limits_{\sqrt {\lambda}}^{\sqrt {2 \lambda}} 2t^2.dt - \lambda \int \limits _{\sqrt {\lambda}} ^{\sqrt {2 \lambda}} \large\frac{1}{t}$$ 2t.dt$
$\large\frac{2}{3} \bigg[t^3\bigg]_{\sqrt{\lambda}} ^{\sqrt {2 \lambda}}$$-2 \lambda \bigg[t \bigg]_{\sqrt{\lambda}} ^{\sqrt {2 \lambda}}$
=> $\large\frac{2}{3}$$ [( 2 \lambda)^3 -(\sqrt {\lambda})^3]- 2 \lambda [ \sqrt {2 \lambda}- \sqrt {\lambda}]$
=> $\large\frac{2}{3}$$ {\lambda }^3 [ ( 2 \sqrt {2}) - 1]- 2 (\lambda)^{3/2} [ \sqrt {2 \lambda}- 1]$
=> $I= \large\frac{2}{3}$$ (2 - \sqrt 2) \lambda \sqrt {\lambda}$
Hence the correct answer is a
answered Dec 17, 2013 by meena.p
 

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