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

$(a)\;\frac{-2}{(p-2) \sqrt {x-p} \sqrt {x-q}}+ (x-p)+c \\(b)\;(x-p)^2+(x-q)^2+c \\(c)\; x^2+c \\ (d)\;None$
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1 Answer

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if $\int \large\frac{dx}{(ax+b)^m (x+q)^n}$
$m+n=2,$ and m,n are differentiate with respect to x
$\large\frac{ax+b}{cx+d}$$=t$
and then solve
=> $\int \large\frac{dx}{(x-p)^{3/2} (x-q)^{1/2}}$
$m+n=2$
=> $\large\frac{x-p}{x-q}$$=t$; differentiate with respect to x
$\large\frac{(x-q)-(x-p)}{(x-q)^2}$$dx$
=> $\large\frac{p-q}{(x-q)^2}$$dx=dt$
=> $\int \large\frac{dx}{\bigg(\Large\frac{x-p}{x-q}\bigg)^{3/2}. (x-q)^2}$
=> $\int \large\frac{1}{t^{3/2}} .\frac{1}{(p-q)}$$.dt$
=> $ \large\frac{1}{p-q}$$ \int t^{-3/2}.dt$
=> $\large\frac{1}{p-q} \frac{(-2)}{\sqrt t} $$+c$
=> $\large\frac{-2}{(p-q)\sqrt {x-p}\sqrt{x-q}}$$+c$
Hence d is the correct answer.
answered Dec 31, 2013 by meena.p
 
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