Question

integration of underroot 9+x÷9-x .dx​

Answer 1

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

Given integral is

$\rm :\longmapsto\:\displaystyle\int\rm \sqrt{ \frac{9 - x}{9 + x} } \: dx$

On rationalizing the denominator, we get

$\rm \:  =  \:  \: \:\displaystyle\int\rm \sqrt{ \frac{9 - x}{9 + x} \times \frac{9 - x}{9 - x} } \: dx$

$\rm \:  =  \:  \: \displaystyle\int\rm \sqrt{ \frac{ {(9 - x)}^{2} }{(9 + x)(9 - x)} } \:dx$

$\rm \:  =  \:  \: \displaystyle\int\rm \frac{9 - x}{ \sqrt{ {9}^{2} - {x}^{2} } } \: dx$

$\rm \:  =  \:  \: \displaystyle\int\rm \frac{9 - x}{ \sqrt{ 81 - {x}^{2} } } \: dx$

$\rm \:  =  \:  \:9 \displaystyle\int\rm \frac{1}{ \sqrt{ 81 - {x}^{2} } } \: dx + \displaystyle\int\rm \frac{ - x}{ \sqrt{81 - {x}^{2} } } \: dx$

We know,

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

So, using this identity, we get

$\rm \:  =  \:  \: 9 \: {sin}^{ - 1}\dfrac{x}{9} + P_1$

where,

$\rm :\longmapsto\:P_1 = \displaystyle\int\rm \frac{ - x}{ \sqrt{81 - {x}^{2} } } \: dx$

$\bf\implies \:\displaystyle\int\bf \sqrt{ \frac{9 - x}{9 + x} } \: dx = 9 \: {sin}^{ - 1}\dfrac{x}{9} + P_1 - - - (1)$

Now, Consider,

$\rm :\longmapsto\:P_1 = \displaystyle\int\rm \frac{ - x}{ \sqrt{81 - {x}^{2} } } \: dx$

To evaluate this integral, we use method of Substitution,

$\red{\rm :\longmapsto\:Put \: \sqrt{81 - {x}^{2} } = y}$

$\red{\rm :\longmapsto\:81 - {x}^{2} = {y}^{2}}$

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

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

So, on substituting these values, we get

$\rm \:  =  \:  \: \displaystyle\int\rm \frac{y \: dy}{y}$

$\rm \:  =  \:  \: \displaystyle\int\rm dy$

$\rm \:  =  \:  \: y \: + \: c$

$\rm \:  =  \:  \: \sqrt{81 - {x}^{2} } \: + \: c$

$\bf\implies \:P_1 = \sqrt{81 - {x}^{2} } \: + \: c$

On substituting the value in equation (1), we get

$\bf\implies \:\displaystyle\int\bf \sqrt{ \frac{9 - x}{9 + x} } \: dx = 9 \: {sin}^{ - 1}\dfrac{x}{9} + \sqrt{81 - {x}^{2} } + c$

Short Cut Trick :-

To evaluate the integral of the form

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

The result is

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