Mathematics

Simplify:-
$$\int {\left( {\dfrac{1}{{\left( {\ln x} \right)}} - \dfrac{1}{{{{\left( {\ln x} \right)}^2}}}} \right)} dx$$


SOLUTION
$$\displaystyle\int \left(\dfrac{1}{\ln x}-\dfrac{1}{(\ln x)^{2}}\right)\ dx$$
Put $$\ln x=t,\dfrac{dx}{x}=dt$$
$$dx=x\ dt$$
$$dx=e^{t}dt$$
$$\displaystyle\int e^{t}\left(\dfrac{1}{t}-\dfrac{1}{t^{2}}\right) dt$$
We know that $$\displaystyle\int e^{x}(f(x)+f(x))dx=e^{x}f(x)+c$$
$$=\dfrac{e^{t}}{t}+c$$
$$=\dfrac{x}{\ln x}+c=\dfrac{x}{\ln x}+c$$
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Subjective Medium Published on 17th 09, 2020
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