Question

\({\rm{ }}\int_{\rm{e}}^{{{\rm{e}}^{\rm{4}}}} {\frac{{{\rm{dx}}}}{{{\rm{x}}\sqrt {{\rm{lnx}}} }}} .\)

Short answer

The definite integral's evaluation is\({\rm{2}}\).

Step-by-step solution

Explanation of solution.   

Substitute out the \({\rm{x }}\) and \({\rm{dx}}\) for \({\rm{u }}\) and \({\rm{du}}\)

\(\begin{aligned}{l}{\rm{a = ln}}{{\rm{e}}^{\rm{4}}}{\rm{ = 4}}\;\;\;{\rm{,b = lne = 1}}\\{\rm{I = }}\int_{\rm{1}}^{\rm{4}} {\frac{{\rm{1}}}{{\sqrt {\rm{u}} }}} {\rm{du}}\\{\rm{I = }}\int_{\rm{1}}^{\rm{4}} {{{\rm{u}}^{{\rm{ - 1/2}}}}} {\rm{du}}{\rm{.}}\end{aligned}\)

Calculation.              

Know that \(\int_a^b {{F^\prime }} (x) = F(b) - F(a)\)

And also \(\int {{x^n}} dx = \frac{{{x^{n + 1}}}}{{n + 1}} + c\)

Intergrate \({\rm{I = }}\left( {{\rm{2}}{{\rm{u}}^{{\rm{1/2}}}}} \right)_{\rm{1}}^{\rm{4}}\)

\(\) \(\)\(\begin{aligned}{l}{\rm{ = 2}}\left( {{{\rm{4}}^{{\rm{1/2}}}}{\rm{ - }}{{\rm{1}}^{{\rm{1/2}}}}} \right)\\{\rm{ = 2(2 - 1)}}\end{aligned}\)

Hence definite integral's \({\rm{ = 2}}{\rm{.}}\)