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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Evaluate the definite integrals\[\int\limits_0^1\frac{dx}{\sqrt{1+x}-\sqrt x}\]

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  • (i)$\int \limits_a^bf(x)dx=F(b)-F(a)+c$
  • $\int x^n dx=[\frac{x^{n+1}}{n+1}]+c$
Given $I=\int\limits_0^1\frac{dx}{\sqrt{1+x}-\sqrt x}$
Let us multiply and divide by its conjujate,
Therefore $I=\int\limits_0^1\frac{dx}{\sqrt{1+x}-\sqrt x} \times \frac{\sqrt {1+x}+\sqrt x}{\sqrt {1+x}+\sqrt x}$
$=\int\limits_0^1\frac{\sqrt {1+x}+\sqrt x}{1+x-x}dx$
$=\int\limits_0^1({\sqrt {1+x}+\sqrt x})dx$
On seperating the terms,
$I=\int\limits_0^1\sqrt {1+x}dx+\int \limits_0^1\sqrt x\;dx$
On integrating we get,
On applying limits,
$=\frac{2}{3} \bigg\{2^{3/2}-1+1 \bigg\}+c$



answered Feb 19, 2013 by meena.p

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