Mathematics

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


ANSWER

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


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
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