Question

please integrate this​

Answer 1

Given Question is

$\sf \: Evaluate : \: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{3 + 3x + {x}^{2} } }$

Basic Concept Used :-

• Method of Completing squares

Identities Used :-

$\boxed{ \red{ \bf \:\: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{ {x}^{2}+ {a}^{2}}} = \:log |x + \sqrt{ {x}^{2} + {a}^{2} } | + c}}$

Solution :-

$\rm :\longmapsto\:\: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{3 + 3x + {x}^{2} } }$

$\sf \: = \: \: \: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{ {x}^{2} + 3x + 3}}$

On adding and Subtracting the square of half of the coefficient of x, we get

$\sf \: = \: \: \: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{ {x}^{2} + 3x + {\bigg(\dfrac{3}{2} \bigg) }^{2} - {\bigg(\dfrac{3}{2} \bigg) }^{2} + 3}}$

$\sf \: = \: \: \: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{{\bigg(x + \dfrac{3}{2} \bigg) }^{2} + 3 - \dfrac{9}{4}}}$

$\sf \: = \: \: \: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{{\bigg(x + \dfrac{3}{2} \bigg) }^{2} +\dfrac{12 - 9}{4}}}$

$\sf \: = \: \: \: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{{\bigg(x + \dfrac{3}{2} \bigg) }^{2} +\dfrac{3}{4}}}$

$\sf \: = \: \: \: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{{\bigg(x + \dfrac{3}{2} \bigg) }^{2} +{\bigg(\dfrac{ \sqrt{3} }{2} \bigg) }^{2}}}$

$\sf \: = \: \: log \bigg |{\bigg(x + \dfrac{3}{2} \bigg) } + \sqrt{{\bigg(x + \dfrac{3}{2} \bigg) }^{2} +{\bigg(\dfrac{ \sqrt{3} }{2} \bigg) }^{2}} \bigg| + c$

$\sf \: = \: \: log \bigg |{\bigg(x + \dfrac{3}{2} \bigg) } + \sqrt{{\bigg( {x}^{2} + \dfrac{9}{4} + 3x \bigg) } +{\bigg(\dfrac{3}{4} \bigg) }} \bigg| + c$

$\sf \: = \: \: log \bigg |{\bigg(x + \dfrac{3}{2} \bigg) } + \sqrt{{{x}^{2} + 3x + 3}} \bigg| + c$

Additional Information :-

$\boxed{ \red{ \bf \:\: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{ {x}^{2} - {a}^{2}}} = \:log |x + \sqrt{ {x}^{2} - {a}^{2} } | + c}}$

$\boxed{ \red{ \bf \:\: \displaystyle \int \sf \: \dfrac{dx}{ \sqrt{ {a}^{2} - {x}^{2} } } = \dfrac{1}{a} {sin}^{ - 1}\dfrac{x}{a} + c}}$

$\boxed{ \red{ \bf \:\: \displaystyle \int \sf \: \dfrac{dx}{ {x}^{2} + {a}^{2} } = \dfrac{1}{a} {tan}^{ - 1}\dfrac{x}{a} + c}}$

$\boxed{ \red{ \bf \:\: \displaystyle \int \sf \: \dfrac{dx}{ {x}^{2} - {a}^{2} } = \dfrac{1}{2a}log |\dfrac{x - a}{x + a} | + c}}$