_ ,1 ^ ,x x^2 x ,dx = - Vidyadip

MCQ

$\int_{\,1}^{\,x} {\frac{{\log {x^2}}}{x}\,dx = } $

✓
${(\log x)^2}$
B
$\frac{1}{2}{(\log x)^2}$
C
$\frac{{\log {x^2}}}{2}$
D
None of these

✓

Answer

Correct option: A.

${(\log x)^2}$

a
(a) $I = \int_1^x {\frac{{2\log x}}{x}dx} $

Let $\log x = t$

==> $\frac{{dx}}{x} = dt$

$\therefore I = 2\int_0^{\log x} {t\,dt = 2\,\left[ {\frac{{{t^2}}}{2}} \right]} _0^{\log x} = {(\log x)^2}$.

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

If angle $\theta $ be divided into two parts such that the tangent of one part is $k$ times the tangent of the other and $\phi $ is their difference, then $\sin \theta = $

If the system of equations $2x + 3y - z = 0$, $x + ky - 2z = 0$ and $2x - y + z = 0$ has a non -trivial solution $(x, y, z)$, then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k$ is equal to

$\tan 3A - \tan 2A - \tan A = $

The number of common tangents to the circles ${x^2} + {y^2} - 4x - 6y - 12 = 0$ and ${x^2} + {y^2} + 6x + 18y + 26 = 0$ is

If the shortest distance between the lines $\vec{r}_{1}=\alpha \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k}), \lambda \in R, \alpha>0$ and $\vec{r}_{2}=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k}), \mu \in R$ is $9$, then $\alpha$ is equal to $.....$

If $\alpha_1<\alpha_2<\alpha_3<\alpha_4<\alpha_5<\alpha_6$, then the equation $(x-\alpha_1)(x-\alpha_3) (x-\alpha_5) + 3 (x-\alpha_2) (x-\alpha_4) (x-\alpha_6) = 0$ has :-

There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves proved to exceed by $66$ the number of games that the men played with the women. The number of participants is

If equation of three sides of a triangle are $x = 2,$ $y + 1 = 0$ and $x + 2y = 4$ then co-ordinates of circumcentre of this triangle are

Let $C$ be the largest circle centred at $(2,0)$ and inscribed in the ellipse $=\frac{x^2}{36}+\frac{y^2}{16}=1$.If $(1, \alpha)$ lies on $C$, then $10 \alpha^2$ is equal to $.........$

$S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x + y + z =1$ ; $x +\lambda y + z =1$ ; $x + y +\lambda z =1$ is inconsistent, then $\sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)$ is equal to