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

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

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

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

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

$\left| {\,\begin{array}{*{20}{c}}{{b^2} + {c^2}}&{{a^2}}&{{a^2}}\\{{b^2}}&{{c^2} + {a^2}}&{{b^2}}\\{{c^2}}&{{c^2}}&{{a^2} + {b^2}}\end{array}\,} \right| = $

If $\alpha $ and $\beta $ are roots of the equation ${x^2} - x + 1 = 0$ then ${\alpha ^{2009}} + {\beta ^{2009}} = $ . . . .

If ${\cos ^{ - 1}}x - {\cos ^{ - 1}}\frac{y}{2} = \alpha $, then $4{x^2} - 4xy\cos \alpha + {y^2}$ is equal to

If $x,y,z$ are in $A.P. $ and ${\tan ^{ - 1}}x,{\tan ^{ - 1}}y$ and ${\tan ^{ - 1}}z$ are also in $A.P.$, then

The number of all possible positive integral values of $\alpha $ for which the roots of the quadratic equation, $6x^2 - 11x +\alpha =0$ are rational numbers is

$\frac{{\sin 3\theta + \sin 5\theta + \sin 7\theta + \sin 9\theta }}{{\cos 3\theta + \cos 5\theta + \cos 7\theta + \cos 9\theta }} = $

Let $D _{ k }=\left|\begin{array}{ccc}1 & 2 k & 2 k -1 \\ n & n ^2+ n +2 & n ^2 \\ n & n ^2+ n & n ^2+ n +2\end{array}\right|$. If $\sum \limits_{ k =1}^n$ $D _{ k }=96$, then $n$ is equal to

If $A$ is a square matrix of $3 \times 3$ order, and $|A| = 2$ then $|(A-A^T)^6| + |(A^T-A)^7|$ is equal to (where $A^T$ donotes the transpose of matrix $A$).

Consider a region $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathrm{R}^{2}: \mathrm{x}^{2} \leq \mathrm{y} \leq 2 \mathrm{x}\right\}$ If a line $\mathrm{y}=\alpha$ divides the area of region $\mathrm{R}$ into two equal parts, then which of the following is true?

The region formed by the inequalities $x, y \geq 0, y \leq 6, x+y \leq 3$ is $.....$