Mathematics

Solve :
$$ \displaystyle \int \dfrac {dy}{e^{(x-1)y} -1 } $$


SOLUTION
Consider, $$I=\displaystyle\int { \dfrac { dy }{ { e }^{ \left( x-1 \right) y }-1 }  }$$

$$\Rightarrow I= \displaystyle\int { \dfrac { dy }{ \dfrac { { e }^{ xy } }{ { e }^{ y } } -1 }  }$$

$$=\displaystyle\int { \dfrac { { e }^{ y }dy }{ { e }^{ xy }-{ e }^{ y } }  }$$

$${ e }^{ y }=t$$ $$\Rightarrow { e }^{ y }dy=dt$$

$$I= \displaystyle\int { \dfrac { dt }{ { e }^{ x }t-t }  }$$

$$= \displaystyle\int { \dfrac { dt }{ t\left( { e }^{ x }-1 \right)  }  }$$

$$= \dfrac { 1 }{ \left( { e }^{ x }-1 \right)  } \displaystyle\int { \dfrac { dt }{ t }  }$$

$$= \dfrac { 1 }{ \left( { e }^{ x }-1 \right)  } \left( n\left| t \right| +\left| n \right| c \right)$$

$$= c{ t }^{ \left( { e }^{ x }-1 \right)  }$$

$$= c{ \left( { e }^{ y } \right)  }^{ { e }^{ x }-1 }$$

$$\boxed{\Rightarrow c{ e }^{ y\left( { e }^{ x }-1 \right)  }}$$
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