$\int x . (x^{x})^{x} . (2 l o g x + 1) d x$ is equal to

x^{(x^{x})} + C

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(x^{x})^{x} + C

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x^{x} . l o g x + C

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None of these

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Solution

The correct option is B $(x^{x})^{x} + C$
$\int x . (x^{x})^{x} . (2 l o g x + 1) d x$
$(x^{x})^{x} = x^{x^{2}}$
$x^{x^{2}} = t$
$x^{2} l o g x = l o g t$
$(2 x l o g x + x) d x = \frac{1}{t} d t$
$x (2 l o g x + 1) d x = \frac{1}{x} d t$
$= \int t . \frac{1}{t} d t$
$= t + c$
$= (x^{x})^{x} + c$

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