INTEGRATION ( A DAS GUPTA)

1- \sqrt{x} = t ^ 2

t = \sqrt{ 1- \sqrt{x}}

\sqrt{x} = 1- t^2

\frac{-1}{2\sqrt{x}}dx = 2 t dt

integral reduces to

\int \frac{ - 2 \sqrt{x} . 2t dt }{t}

- 4\int \sqrt{x} dt = - 4\int \left(1- t^ 2 \right) dt

= - 4t + \frac{4t^ 3}{3} + c

- 4 \sqrt{1-\sqrt{x}} + 4(\sqrt{1-\sqrt{x}}) ^{3} /3

take \sqrt{1-\sqrt{x}}

as common u get

-\frac{4}{3}\sqrt{1-\sqrt{x}}\left(3 - \sqrt{(1 - \sqrt{x}) ^2} \right)

-\frac{4}{3}\sqrt{1-\sqrt{x}}\left(2 + \sqrt{x} \right)

thnk u.

put\;x = (1-m^{2})^{2}

substitute \sqrt{x}=t

2\int \frac{t}{\sqrt{1-t}}dt=2\int t{(1-t)^{-1/2}}dt

now substitue 1-t=m²