Question

integrate this function​

Answer 1

$\large\underline{\sf{Solution-}}$

The given integral is

$\rm :\longmapsto\:\displaystyle\int\tt \frac{dx}{ \sqrt{x + 1} - \sqrt{x} }$

To evaluate this integral, we use method of Rationalization.

$\rm \: = \: \: \displaystyle\int\tt \frac{1}{ \sqrt{x + 1} - \sqrt{x} } \times \frac{ \sqrt{x + 1} + \sqrt{x} }{ \sqrt{x + 1} + \sqrt{x} } \: dx$

$\rm \: = \: \: \displaystyle\int\tt \frac{ \sqrt{x + 1} + \sqrt{x} }{( \sqrt{x + 1} - \sqrt{x} )( \sqrt{x + 1} + \sqrt{x})} \: dx$

We know that

$\boxed{ \rm{ (x + y)(x - y) = {x}^{2} - {y}^{2}}}$

So, using this,

$\rm \: = \: \: \displaystyle\int\tt \frac{ \sqrt{x + 1} + \sqrt{x} }{( \sqrt{x + 1})^{2} - (\sqrt{x} )^{2} } \: dx$

$\rm \: = \: \: \displaystyle\int\tt \frac{ \sqrt{x + 1} + \sqrt{x} }{x + 1 - x } \: dx$

$\rm \: = \: \: \displaystyle\int\tt \frac{ \sqrt{x + 1} + \sqrt{x} }{1} \: dx$

$\rm \: = \: \: \displaystyle\int\tt \sqrt{x + 1}dx + \displaystyle\int\tt \sqrt{x} dx$

$\rm \: = \: \: \displaystyle\int\tt {\bigg((x + 1) \bigg) }^{\dfrac{1}{2} } dx+ \displaystyle\int\tt {\bigg(x \bigg) }^{\dfrac{1}{2} }dx$

We know,

$\boxed{ \rm{ \displaystyle\int\tt {x}^{n} \: dx \: = \frac{ {x}^{n + 1} }{n + 1} + c}}$

$\rm \: = \: \: \dfrac{{\bigg(x + 1 \bigg) }^{\dfrac{1}{2} + 1}}{\dfrac{1}{2} + 1} + \dfrac{{\bigg(x \bigg) }^{\dfrac{1}{2} + 1}}{\dfrac{1}{2} + 1} + c$

$\rm \: = \: \: \dfrac{{\bigg(x + 1 \bigg) }^{\dfrac{3}{2}}}{\dfrac{3}{2}} + \dfrac{{\bigg(x \bigg) }^{\dfrac{3}{2}}}{\dfrac{3}{2}} + c$

$\rm \: = \: \: \dfrac{2}{3}{\bigg(x + 1\bigg) }^{\dfrac{3}{2} } + \dfrac{2}{3}{\bigg(x\bigg) }^{\dfrac{3}{2} } + c$

Additional Information :-

$\boxed{ \rm{ \displaystyle\int\tt sinx \: dx \: = - \: cosx + c}}$

$\boxed{ \rm{ \displaystyle\int\tt cosx \: dx \: = \: sinx + c}}$

$\boxed{ \rm{ \displaystyle\int\tt cosecx cotx \: dx \: = \: - \: cosecx + c}}$

$\boxed{ \rm{ \displaystyle\int\tt secx tanx \: dx \: = \: \: secx + c}}$

$\boxed{ \rm{ \displaystyle\int\tt {sec}^{2}x \: dx \: = \: tanx + c}}$

$\boxed{ \rm{ \displaystyle\int\tt {cosec}^{2}x \: dx \: = - \: cotx + c}}$