Mathematics

# Evaluate: $\int _{ 3 }^{ 9 }{ \cfrac { \sqrt [ 3 ]{ 12-x } }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ 12-x } } } dx$

##### SOLUTION
$I=\displaystyle\int _{ 3 }^{ 9 }{ \cfrac { \sqrt [ 3 ]{ 12-x } }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ 12-x } } } dx=\int_{3}^{9}{\cfrac{\sqrt[3]{12-(9+3-x)}}{\sqrt[3]{9+3-x}+\sqrt[3]{12-(9+3-x)}}}dx$

$I=\displaystyle\int _{ 3 }^{ 9 }{ \cfrac { \sqrt [ 3 ]{ 12-x } }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ 12-x } } } dx=\int_{3}^{9}{\cfrac{\sqrt[3]{x}}{\sqrt[3]{12-x}+\sqrt[3]{x}}}dx$

$2I=\displaystyle\int _{ 3 }^{ 9 }{ \cfrac { \sqrt [ 3 ]{ 12-x } }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ 12-x } } } dx+\int_{3}^{9}{\cfrac{\sqrt[3]{x}}{\sqrt[3]{12-x}+\sqrt[3]{x}}}dx$

$2I=\displaystyle\int_{3}^{9}dx=6\implies I=3$

$\displaystyle\int _{ 3 }^{ 9 }{ \cfrac { \sqrt [ 3 ]{ 12-x } }{ \sqrt [ 3 ]{ x } +\sqrt [ 3 ]{ 12-x } } } dx=3$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

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