Mathematics

$$\displaystyle\int \frac{3x^{2}}{\sqrt{\left ( 9-16x^{6} \right )}}dx.$$


ANSWER

$$\displaystyle \frac{1}{4}\sin ^{-1}\frac{4}{3}x^{3.}$$


SOLUTION
Let $$\displaystyle I=\int  \frac { 3x^{ 2 } }{ \sqrt { \left( 9-16x^{ 6 } \right)  }  } dx$$

Put $$\displaystyle 4x^{ 3 }=t\Rightarrow 12x^{ 2 }dx=dt$$
Therefore
$$\displaystyle I=\frac { 1 }{ 4 } \int  \frac { dt }{ \sqrt { \left( 3^{ 2 }-t^{ 2 } \right)  }  } =\frac { 1 }{ 4 } \sin ^{ -1 } \frac { t }{ 3 } =\frac { 1 }{ 4 } \sin ^{ -1 } \frac { 4 }{ 3 } x^{ 3 }$$
Hence, option 'D' is correct.
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Single Correct Medium Published on 17th 09, 2020
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