Question

Evaluate:-$\huge \displaystyle \sf \int \dfrac{ \sqrt{x^{2} + a^{2} } }{x} dx$​

Answer 1

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

The given integral is

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

can be rewritten as

$\rm \: = \: \: \displaystyle \rm \int \dfrac{ \sqrt{x^{2} + a^{2} } }{ {x}^{2} } \: x \: dx$

To evaluate this integral, we use method of Substitution,

$\red{\rm :\longmapsto\:Put \: \sqrt{ {x}^{2} + {a}^{2} } = y}$

On squaring both sides, we get

$\red{\rm :\longmapsto \: {x}^{2} + {a}^{2} = {y}^{2} }$

$\red{\rm :\longmapsto \: {2x}dx = 2{y}dy}$

$\red{\rm :\longmapsto \: {x}dx = {y}dy}$

So, on substituting the values, we have

$\rm \: = \: \: \displaystyle \rm \int \dfrac{ y }{ {y}^{2} - {a}^{2} } \: y \: dy$

$\rm \: = \: \: \displaystyle \rm \int \dfrac{ {y}^{2} }{ {y}^{2} - {a}^{2} } \: dy$

$\rm \: = \: \: \displaystyle \rm \int \dfrac{ {y}^{2} - {a}^{2} + {a}^{2} }{ {y}^{2} - {a}^{2} } \: dy$

$\rm \: = \: \: \displaystyle \rm \int \dfrac{ {y}^{2} - {a}^{2}}{ {y}^{2} - {a}^{2} } \: dy \: + \: \displaystyle \rm \int \dfrac{{a}^{2}}{ {y}^{2} - {a}^{2} } \: dy$

$\rm \: = \: \: \displaystyle \rm \int 1\: dy \: + {a}^{2} \: \displaystyle \rm \int \dfrac{1}{ {y}^{2} - {a}^{2} } \: dy$

$\rm \: = \: \: y + {a}^{2}\dfrac{1}{2a} log \bigg |\dfrac{y - a}{y + a} \bigg| + c$

$\rm \: = \: \: y + \dfrac{a}{2} log \bigg |\dfrac{y - a}{y + a} \bigg| + c$

$\rm \: = \: \: \sqrt{ {x}^{2} + {a}^{2} } + \dfrac{a}{2} log \bigg |\dfrac{ \sqrt{ {x}^{2} + {a}^{2} } - a}{ \sqrt{ {x}^{2} + {a}^{2} } + a} \bigg| + c$

Additional Information :-

$\boxed{ \rm \: \displaystyle \rm \int \frac{dx}{ {x}^{2} +{a}^{2}} = \frac{1}{a} {tan}^{ -1 } \frac{x}{a} \: + \: c}$

$\boxed{ \rm \: \displaystyle \rm \int \frac{dx}{ \sqrt{{x}^{2} +{a}^{2}}} = log |x + \sqrt{ {x}^{2} + {a}^{2} } | \: + \: c }$

$\boxed{ \rm \: \displaystyle \rm \int \frac{dx}{ \sqrt{{x}^{2} - {a}^{2}}} = log |x + \sqrt{ {x}^{2} - {a}^{2} } | \: + \: c }$

$\boxed{ \rm \: \displaystyle \rm \int \frac{dx}{ \sqrt{ {a}^{2} - {x}^{2} } } = {sin}^{ - 1} \frac{x}{a} \: + \: c}$

$\boxed{ \rm \: \displaystyle \rm \int \frac{dx}{ {x}^{2} - {a}^{2} } = \dfrac{1}{2a} log \bigg |\dfrac{x - a}{x + a} \bigg| + c }$

$\boxed{ \rm \: \displaystyle \rm \int \frac{dx}{ {a}^{2} - {x}^{2} } = \dfrac{1}{2a} log \bigg |\dfrac{a + x}{a - x} \bigg| + c }$