₀¹{dx}{{1+x}-{x}} - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Evaluate the definite integrals in Exercises 253 to 261.

Question : 257 of 271

Marks: +1, -0

\int\limits_{0}^{1}\frac{dx}{\sqrt{1+x}-\sqrt{x}}

Solution:

Let

I=\int\limits_{0}^{1}\frac{dx}{\sqrt{1+x}-\sqrt{x}}

=\int\limits_{0}^{1}\frac{\sqrt{1+x}-\sqrt{x}}{(1+x)-x}\,dx

=\int\limits_{0}^{1}\left[\sqrt{1+x}-\sqrt{x}\right]\,dx

=\left[\frac{2}{3}(1+x)^{3/2}\right]_{0}^{1}+\left[\frac{2}{3}x^{3/2}\right]_{0}^{1}

=\frac{2}{3}(2^{3/2}-1)+\frac{2}{3}

=\frac{2}{3}\cdot 2^{3/2}-\frac{2}{3}+\frac{2}{3}

=\frac{2}{3}\cdot 2\sqrt{2}=\frac{4\sqrt{2}}{3}

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