The integral _ 1 ^ 2 e ^ x x ^ x (2+ _ e x ) d x equal - Vidyadip

MCQ

The integral $\int \limits_{1}^{2} e ^{ x } \cdot x ^{ x }\left(2+\log _{ e } x \right) d x$ equal

A
$e (4 e +1)$
B
$e(2 e-1)$
C
$4 e^{2}-1$
✓
$e (4 e -1)$

✓

Answer

Correct option: D.

$e (4 e -1)$

d
$\int_{1}^{2} e ^{x} \cdot x ^{ x }\left(2+\log _{ e } x \right) d x$

$\int_{1}^{2} e ^{ x }\left(2 x ^{ x }+ x ^{ x } \log _{ e } x \right) d x$

$\int_{1}^{2} e ^{ x }(\frac{ x ^{ x }}{f( x )}+\underbrace{ x ^{ x }\left(1+\log _{ e } x \right)}_{f^{(}( x )}) d x$

$\left( e ^{ x } \cdot x ^{ x }\right)_{1}^{2}=4 e ^{2}- e$

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