Mathematics

Evaluate: $$I=\int \dfrac{e^{x}}{(1+e^{x})(2+e^{x})}$$ dx


SOLUTION
$$\begin{array}{l} \int _{  }^{  }{ \frac { { { e^{ x } } } }{ { \left( { 1+{ e^{ x } } } \right) \left( { x+{ e^{ x } } } \right)  } }  } dx \\ Let, \\ 1+{ e^{ x } }=t \\ \frac { { dt } }{ { dx } } ={ e^{ x } } \\ =\int _{  }^{  }{ \frac { { dt } }{ { t\left( { t+1 } \right)  } }  }  \\ =\int _{  }^{  }{ \frac { { dt } }{ { { t^{ 2 } }+t } }  }  \\ =\int _{  }^{  }{ \frac { { dt } }{ { { { \left( { t+\frac { 1 }{ 2 }  } \right)  }^{ 2 } }-{ { \left( { \frac { 1 }{ 2 }  } \right)  }^{ 2 } } } }  }  \\ =\frac { 1 }{ { 2\times \frac { 1 }{ 2 }  } } \ln { \left| { \frac { { \left( { t+\frac { 1 }{ 2 }  } \right) -\frac { 1 }{ 2 }  } }{ { \left( { t+\frac { 1 }{ 2 }  } \right) +\frac { 1 }{ 2 }  } }  } \right|  } +C \\ =\ln { \left| { \frac { t }{ { t+1 } }  } \right|  } +C \\ I=\ln { \left| { \frac { { 1+{ e^{ x } } } }{ { 2+{ e^{ x } } } }  } \right|  } +C \end{array}$$
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Subjective Medium Published on 17th 09, 2020
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