MCQ
The area bounded by the curves $y = {\log _e}x$ and $y = {({\log _e}x)^2}$ is
- ✓$3 - e$
- B$e - 3$
- C$\frac{1}{2}(3 - e)$
- D$\frac{1}{2}(e - 3)$
✓
Answer
Correct option: A.
$3 - e$
a
(a) Required area $ = \int_1^e {[\log x - {{(\log x)}^2}]} \,dx$
(a) Required area $ = \int_1^e {[\log x - {{(\log x)}^2}]} \,dx$
$A = \int_1^e {\,\log x\,dx} - \int_1^e {{{(\log x)}^2}dx} $
$ = [x\log x - x]\,_1^e - [x{(\log x)^2} - 2x\log x + 2x]\,_1^e$
$ = [e - e - ( - 1)] - [e{(1)^2} - 2e + 2e - (2)]$
$ = (1) - (e - 2)$$ = 3 - e$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
For the system of linear equations :
→$x-2 y=1, x-y+k z=-2, k y+4 z=6, k \in R$
consider the following statements :
$(A)$ The system has unique solution if $k \neq 2$, $k \neq-2$
$(B)$ The system has unique solution if $k =-2$.
$(C)$ The system has unique solution if $k =2$.
$(D)$ The system has no-solution if $k =2$.
$(E)$ The system has infinite number of solutions if $k \neq-2$
Which of the following statements are correct?
A cubical die faces marked $1 ,2,3, ... ,6$ is tossed such that the probability of throwing the number $t$ is proportional to $t^2$. The probability that the number $5$ has appeared given that when the die is rolled the number turned up is not even, is
→If $|a|\,\, = 3,\,\,\,|b\,\,|\,\, = 4$ and the angle between $ a$ and $b$ be ${120^o}$, then $|4a + 3b|\,\, = $
→$\int_{}^{} {{x^2}\sin 2x} \;dx = $
→A function $y = f(x)$ satisfies the differential equation $f(x).sin\ 2x\ -\ cos\ x\ +\ (1 + sin^2x) f'(x) = 0$ where $f(0) = 0$ . Then value of $f(\frac {\pi}{6})$ is equal to
→If the integral $\int {\frac{{\cos \,8x + 1}}{{\cot \,2x - \tan \,2x}}} dx = A\,\cos \,8x + k,$ where $k$ is an arbitrary constant, then $A$ is equal to
→If $\mathrm{A}=\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right],$ then verify that $\mathrm{A}$ $\mathrm{adj}$ $\mathrm{A}=| \mathrm{A} | \mathrm{I}$. Also find $\mathrm{A}^{-1}$.
→Suppose that two cards are drawn at random from a deck of cards. Let $X$ be the number of aces obtained. Then the value of $\mathrm{E}(\mathrm{X})$ is
→The vectors $\overrightarrow {AB\,} \, = \,3\hat i\, + \,5\hat j\, + \,\,4\hat k\,\,and\,\,\overrightarrow {AC} \, = \,5\hat i\, - 5\hat j\, + 2\hat k$ are the sides of a triangle $ABC.$ The length of the median through $A$ is .............. $\mathrm{unit}$
→The area of the region enclosed by the parabola x2 = y, the line y = x + 2 and the x - axis, is:
→- $\frac{2}{9}$
- $\frac{9}{2}$
- $9$
- $2$