Mathematics

$\displaystyle \int \frac{x}{\sqrt{\left ( 9+8x-x^{2} \right )}}dx.$

$\displaystyle -\sqrt{9+8x-x^{2}}+\sin ^{-1}\frac{x-4}{5}$

SOLUTION
Let $\displaystyle I=\int \frac { x }{ \sqrt { \left( 9+8x-x^{ 2 } \right) } } dx$

Comparing $\displaystyle x=l(8-2x)+m\Rightarrow -2l=1,8l+m=0\Rightarrow I=-\frac { 1 }{ 2 } ,m=4$

Therefore

$\displaystyle I=l\int \frac { 8-2x }{ \sqrt { \left( 9+8x-x^{ 2 } \right) } } dx+m\int \frac { dx }{ \sqrt { \left[ 25-\left( x^{ 2 }-8x+16 \right) \right] } }$

Put $9+8x-x^{ 2 }=t\Rightarrow \left( 8-2x \right) dx=dt$

Therefore

$\displaystyle I=l\int \frac { 1 }{ \sqrt { \left( t \right) } } dt+m\int \frac { dx }{ \sqrt { \left[ 5^{ 2 }-\left( x-4 \right) ^{ 2 } \right] } }$

$\displaystyle =l\times2\sqrt { t } +m\sin ^{ -1 } \frac { x-4 }{ 5 } =-\sqrt { 9+8x-x^{ 2 } } +\sin ^{ -1 } \frac { x-4 }{ 5 }$

Hence, option 'C' is correct.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

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