Mathematics

If $$\int \frac { 1 } { 5 + 4 \cos 2 \theta } d \theta = A \tan ^ { - 1 } ( B \tan \theta ) + c$$ then $$( A , B ) =$$


SOLUTION
$$\begin{array}{l} \int { \frac { { d\theta  } }{ { 5+4\cos  2\theta  } } =\int { \frac { { d\theta  } }{ { 5+\frac { { 4\left( { 1-\tan  \theta  } \right)  } }{ { 1+{ { \tan   }^{ 2 } }\theta  } }  } }  }  }  \\ =\int { \frac { { { { \sec   }^{ 2 } }\theta d\theta  } }{ { 9+{ { \tan   }^{ 2 } }\theta  } }  }  \\ Let\, \tan  \theta =u \\ \frac { { du } }{ { d\theta  } } ={ \sec ^{ 2 }  }\theta  \\ Now, \\ =\int { \frac { { du } }{ { 9+{ u^{ 2 } } } } =\frac { 1 }{ 3 } { { \tan   }^{ -1 } }\left( { \frac { { \tan  \theta  } }{ 3 }  } \right) +C }  \\ \left( { A\, ,B } \right) =\left( { \frac { 1 }{ 3 } ,\, \frac { 1 }{ 3 }  } \right)  \end{array}$$
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Subjective Medium Published on 17th 09, 2020
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