Mathematics

If $$\displaystyle \int \dfrac{x + \sqrt[3]{x^2} + \sqrt[6]{x}}{x(1+\sqrt[3]{x})} dx = ax^{\frac{2}{3}} + b\ \tan^{-1}(\sqrt[6]{x}) +c $$ then:


ANSWER

$$ a= \frac{3}{2}$$


SOLUTION
$$\int { \dfrac { x+{ x }^{ 2/3 }+{ x }^{ 1/6 } }{ x\left( { x }^{ 1/3 }+1 \right)  }  } dx$$
$$u={ x }^{ 1/6 }$$
$$du=\dfrac { 1 }{ 6 } \dfrac { dx }{ { x }^{ 5/6 } } $$
$${ x }^{ 1/3 }={ u }^{ 2 }$$
$${ x }^{ 2/3 }={ u }^{ 4 }$$
$$x={ u }^{ 6 }$$
$$x={ u }^{ 6 }$$
$$=6\int { \dfrac { { u }^{ 5 }+{ u }^{ 3 }+1 }{ { u }^{ 2 }+1 }  } du$$
Polynomial division
$$\Rightarrow 6\int { \left( \dfrac { 1 }{ { u }^{ 2 }+1 } +{ u }^{ 3 } \right)  } du$$
$$\Rightarrow 6{ \tan }^{ -1 }\left( u \right) +\dfrac { { 6u }^{ 4 } }{ 4 } +c$$
$$=6{ \tan }^{ -1 }\left( { x }^{ 1/6 } \right) +\dfrac { { 3x }^{ 2/3 } }{ 2 } +c$$
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Single Correct Hard Published on 17th 09, 2020
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