Let y=y(x) be the solution of the differential equation d y d x +… - Vidyadip

MCQ

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+\frac{2 x}{\left(1+x^2\right)^2} y=x e^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0$. Then the area enclosed by the curve $f(\mathrm{x})=\mathrm{y}(\mathrm{x}) \mathrm{e}^{-\frac{1}{\left(1+\mathrm{x}^2\right)}}$ and the line $\mathrm{y}-\mathrm{x}=4$ is....................

A
$62$
✓
$18$
C
$35$
D
$16$

✓

Answer

Correct option: B.

$18$

b
$ \text { IF }=e^{\int \frac{2 x}{\left(1+x^2\right)^2} d x}=e^{\frac{-1}{1+x^2}} $

$ y \cdot e^{\frac{-1}{1+x^2}}=\int x \cdot e^{\frac{1}{1+x^2}} \cdot e^{\frac{-1}{1+x^2} d x} $

$ y \cdot e^{\frac{-1}{1+x^2}}=\frac{x^2}{2}+c $

$ (0,0) \Rightarrow C=0 $

$ y(x)=\frac{x^2}{2} e^{\frac{1}{1+x^2}} $

$ f(x)=\frac{x^2}{2}$

$Imsge$

$A=\int_{-2}^4(x+4)-\frac{x^2}{2} d x=18$

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

The equation of the circle having centre $(1,\; - 2)$ and passing through the point of intersection of lines $3x + y = 14$, $2x + 5y = 18$ is

If $f(x) = \log \left[ {\frac{{1 + x}}{{1 - x}}} \right]$, then $f\left[ {\frac{{2x}}{{1 + {x^2}}}} \right]$ is equal to

Let $S$ be the focus of $ y^2 = 4x $ and a point $P$ is moving on the curve such that it's abscissa is increasing at the rate of $4$ units/sec, then the rate of increase of projection of $SP$ on $x + y = 1$ when $P$ is at $(4, 4)$ is

The number of ways in which $5$ male and $2$ female members of a committee can be seated around a round table so that the two female are not seated together is

If $[m\ n]\left[ {\begin{array}{*{20}{c}}m\\n\end{array}} \right] = [25]$ and $m< n$, then $(m, n) =$

If $x + y + z = {180^o},$ then $\cos 2x + \cos 2y - \cos 2z$ is equal to

The equation to the chord joining two points $(x_1, y_1)$ and $(x_2, y_2)$ on the rectangular hyperbola $xy = c^2$ is

If $f(x) = \left\{ \begin{array}{l}\,\,\,\,{x^2},\,{\rm{when}}\,\,x \le 1\\x + 5,{\rm{when\,\, }}x > {\rm{1}}\end{array} \right.$, then

The line $3x + 2y = 24$ meets $y$-axis at $A$ and $x$-axis at $B$. The perpendicular bisector of $AB$ meets the line through $(0, - 1)$ parallel to $x$-axis at $C$. The area of the triangle $ABC$ is ............... $\mathrm{sq. \, units}$

If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x}{1}=\frac{y}{\alpha}=\frac{z-5}{1}$ is $\frac{5}{\sqrt{6}}$, then the sum of all possible values of $\alpha$ is