Mathematics

Solve $$\int\limits_l^e {\dfrac{{dx}}{{\ln \left( {{x^x} \cdot {e^x}} \right)}}} $$ 


ANSWER

$$\ln 2$$


SOLUTION

Consider the given integral.

$$ I=\int_{1}^{e}{\dfrac{dx}{\ln \left( {{x}^{x}}\cdot {{e}^{x}} \right)}} $$

$$ I=\int_{1}^{e}{\dfrac{dx}{\ln {{x}^{x}}+\ln {{e}^{x}}}} $$

$$ I=\int_{1}^{e}{\dfrac{dx}{x\ln x+x\ln e}} $$

$$ I=\int_{1}^{e}{\dfrac{dx}{x\ln x+x}} $$

$$ I=\int_{1}^{e}{\dfrac{dx}{x\left( \ln x+1 \right)}} $$

 

Let $$t=\ln x+1$$

$$ \dfrac{dt}{dx}=\dfrac{1}{x}+0 $$

$$ dt=\dfrac{dx}{x} $$

 

Therefore,

$$ I=\int_{1}^{2}{\dfrac{dt}{t}} $$

$$ I=\left[ \ln \left( t \right) \right]_{1}^{2} $$

$$ I=\ln \left( 2 \right)-\ln \left( 1 \right) $$

$$ I=\ln 2 $$

 

Hence, this is the answer.

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Single Correct Medium Published on 17th 09, 2020
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