Mathematics

Evaluate $$\displaystyle \int _{ 0 }^{ 2/3 }{ \frac { dx }{ 4+{ 9x }^{ 2 } } } $$


SOLUTION
$$\quad \int _{ 0 }^{ \frac { 2 }{ 3 }  }{ \frac { dx }{ 4+9{ x }^{ 2 } }  } \\ =\frac { 1 }{ 9 } \int _{ 0 }^{ \frac { 2 }{ 3 }  }{ \frac { dx }{ \frac { 4 }{ 9 } +{ x }^{ 2 } }  } \\ =\frac { 1 }{ 9\times \frac { 2 }{ 3 }  } \times { \left[ \tan ^{ -1 }{ \frac { x }{ \frac { 2 }{ 3 }  }  }  \right]  }_{ 0 }^{ \frac { 2 }{ 3 }  }\\ =\frac { 1 }{ 6 } \times \left[ \frac { \pi  }{ 4 }  \right] \\ =\frac { \pi  }{ 24 } $$
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Subjective Medium Published on 17th 09, 2020
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