Mathematics

Solve  $$\int { \cfrac { 1 }{ 9{ x }^{ 2 }+6x+10 } dx } $$


SOLUTION
$$\int { \cfrac { 1 }{ 9{ x }^{ 2 }+6x+10 } dx } =\int { \cfrac { 1 }{ 9{ x }^{ 2 }+6x+1+9 } dx } $$
$$=\int { \cfrac { 1 }{ { (3x+1) }^{ 2 }+9 } dx }$$
$$=\cfrac { 1 }{ 9 } \int { \cfrac { 1 }{ { \left( x+\cfrac { 1 }{ 3 }  \right)  }^{ 2 }+1 } dx } $$
$$=\cfrac { 1 }{ 9 } \int { \cfrac { 1 }{ { 1+\left( x+\cfrac { 1 }{ 3 }  \right)  }^{ 2 } } d\left( x+\cfrac { 1 }{ 3 }  \right)  } $$
$$=\cfrac { 1 }{ 9 } \tan ^{ -1 }{ \left( x+\cfrac { 1 }{ 3 }  \right)  } +C$$
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Subjective Medium Published on 17th 09, 2020
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