Mathematics

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

$\ln 2$

##### 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,

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

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

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

$I=\ln 2$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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