Let the function f :[0,2] R be defined as f(x)= cc e^ [x^2, x-[x]… - Vidyadip

MCQ

Let the function $f :[0,2] \rightarrow R$ be defined as

$f(x)=\left\{\begin{array}{cc}e^{\min \left[x^2, x-[x]\right\}}, & x \in[0,1) \\e^{\left[x-\log _e x\right]}, & x \in[1,2]\end{array}\right.$

where [t] denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int \limits_0^2 x f(x) d x$ is

A
$2 e -1$
B
$1+\frac{3 e }{2}$
✓
$2 e -\frac{1}{2}$
D
$(e-1)\left(e^2+\frac{1}{2}\right)$

✓

Answer

Correct option: C.

$2 e -\frac{1}{2}$

c
Minimum

$m \left\{ x ^2,\{ x \}\right\}= x ^2 ; x \in[0,1)$

${\left[ x -\log _{ e } x \right]=1 ; x \in[1,2)}$

$\therefore f ( x )=\left\{\begin{array}{l} e ^{ x ^2} ; x \in[0,1) \\ e ; x \in[1,2)\end{array}\right.$

$\int \limits_0^2 x f(x) d x=\int \limits_0^1 x e^{x^2} d x+\int \limits_1^2 e x d x$

$=\frac{1}{2}( e -1)+\frac{1}{2}(4-1) e$

$=2 e -\frac{1}{2}$

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

More "M.C.Q (1 Marks)" from STD 12 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

The shortest distance between the lines $\frac{{x - 3}}{2} = \frac{{y + 15}}{{ - 7}} = \frac{{z - 9}}{5}$ and $\frac{{x + 1}}{2} = \frac{{y - 1}}{1} = \frac{{z - 9}}{{ - 3}}$ is

A particle acted on by two forces $3i + 2j - 3k$ and $2i + 4j + 2k$ is displaced from the point $i + 2j + k$ to $5i + 4j + 2k.$ The total work done by the forces is equal to ............ $\mathrm{unit}$

The value of $\int\limits_0^1 {9{x^8}dx + \int\limits_0^{\pi /2} {\cos \,x\,dx} } $ is

If $\text{P(A)}=\frac{3}{10},\text{P(B)}=\frac{2}{5}$ and $\text{P}(\text{A}\cap\text{B}=\frac{3}{5,}$ then P(A|B) + P(B|A) equals

Let $\times $ be a binary operation on set $Q - {1}$ defind by $a \times b = a + b - ab : a, b \in Q - {1}.$ Then $\times $ is:

If ${\tan ^{ - 1}}\frac{{x - 1}}{{x + 2}} + {\tan ^{ - 1}}\frac{{x + 1}}{{x + 2}} = \frac{\pi }{4}$, then $x =$

Let $\beta(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n>0$. If $\int_0^1\left(1-x^{10}\right)^{20} d x=a \times \beta(b, c)$, then $100(a+b+x)$ equals

Let $\alpha \in(0, \infty)$ and $A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$. If $\operatorname{det}\left(\operatorname{adj}\left(2 \mathrm{~A}-\mathrm{A}^{\mathrm{T}}\right) \cdot \operatorname{adj}\left(\mathrm{A}-2 \mathrm{~A}^{\mathrm{T}}\right)\right)=2^8$, then $(\operatorname{det}(\mathrm{A}))^2$ is equal to :

What is the value of $x$ and $y,$ if $2i + 3j = xi + yj:$

Let $y=y(x)$ be the solution curve of the differential equation $\frac{ dy }{ dx }+\frac{1}{ x ^{2}-1} y =\left(\frac{ x -1}{ x +1}\right)^{\frac{1}{2}}$, $x>1$ passing through the point $\left(2, \sqrt{\frac{1}{3}}\right)$. Then $\sqrt{7} y (8)$ is equal to.