Download App

STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
The value of the definite integral,$\int\limits_1^\infty  {{{({e^{x + 1}} + {e^{3 - x}})}^{ - 1}}\,dx} $ is
  • ✓
    $\frac{\pi }{{4{e^2}}}$
  • B
    $\frac{\pi }{{4e}}$
  • C
    $\frac{1}{{{e^2}}}\left( {\frac{\pi }{2} - {{\tan }^{ - 1}}\frac{1}{e}} \right)$
  • D
    $\frac{\pi }{{2{e^2}}}$
✓

Answer

Correct option: A.
$\frac{\pi }{{4{e^2}}}$
a
$I$ =$\int\limits_1^\infty  {\frac{{dx}}{{(e\,\cdot\,{e^x} + {e^3}\,\cdot\,{e^{ - x}})}}} $ =$\int\limits_1^\infty  {\frac{{{e^x}\,dx}}{{e({e^{2x}} + {e^2})}}} $  (multiply $N^r $ and $D^r $ by $e^x$ )
put $e^x = t$ $\Rightarrow $ $e^x dx = dt$
$I =$  $\frac{1}{e}\,\,\int\limits_e^\infty  {\frac{{dt}}{{{t^2} + {e^2}}}} $= $\left. {\frac{1}{{{e^2}}}{{\tan }^{ - 1}}\frac{t}{e}} \right|_e^\infty $ =$\frac{1}{{{e^2}}}\left[ {\frac{\pi }{2} - \frac{\pi }{4}} \right]$ = $\frac{\pi }{{4{e^2}}}$

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 component of $i + j$ along $j + k$ will be
→
If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ and $\vec{\text{d}}$ are the position vector of points $\text{A, B, C, D}$ such that no three of them are collinear and $\vec{\text{a}}+\vec{\text{c}}=\vec{\text{b}}+\vec{\text{d}}$, then $\text{ABCD}$ is a,
→
The sine and cosine curves intersects infinitely many times giving bounded regions of equal areas. The area of one of such region is
→
If $y$ = $|cos 4x| + |sin 4x| + |tan 4x|$ then $\frac{{dy}}{{dx}}$ at $x = \frac{\pi }{6}$ is
→
Evaluate $\begin{bmatrix}3&-1&3\\6&-5&4\\3&-2&3\end{bmatrix}$ is:
→
Let $S$ be the region bounded by the curves $y=x^{3}$ and $y ^{2}= x$. The curve $y =2| x |$ divides $S$ into two regions of areas $R_{1}$ and $R_{2}$.

If $\max \left\{R_{1}, R_{2}\right\}=R_{2}$, then $\frac{R_{2}}{R_{1}}$ is equal to

→
Choose the correct answer from the given four options.
If $A$ and $B$ are matrices of same order, then $(AB\ ' – BA\ ')$ is a :
→
Let $(\mathrm{x}, \mathrm{y})$ be such that

$\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2} .$

Match the statements in Column $I$ with the statements in Column $II$ and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the $ORS$.

Column $I$ Column $II$
$(A)$ If $a=1$ and $b=0$, then ( $x, y$ ) $(p)$ lies on the circle $x^2+y^2=1$
$(B)$ If $a=1$ and $b=1$, then $(x, y)$ $(q)$ lies on $\left(x^2-1\right)\left(y^2-1\right)=0$
$(C)$ If $a=1$ and $b=2$, then ( $x, y)$ $(r)$ lies on $y=x$
$(D)$ If $a=2$ and $b=2$, then $(x, y)$ $(s)$ lies on $\left(4 x^2-1\right)\left(y^2-1\right)=0$
→
The solution to the differential equation $y\, \ln y + xy' = 0,$ where $y (1) = e,$ is
→
$\int {{x^x}(1 + \log x)\,\,dx} $ is equal to
→