Let A = _0^1 , e^t , , ,d ,t 1 , , + , ,t then _ a - 1 ^a , , e^ … - Vidyadip

MCQ

Let $A = \int\limits_0^1 \, \frac{{{e^t}\,\,\,d\,t}}{{1\,\, + \,\,t}}$ then $\int\limits_{a - 1}^a {\,\,\frac{{{e^{ - t}}\,dt}}{{t\, - \,a\, - \,1}}} $ has the value

A
$Ae^{-a}$
✓
$-Ae^{-a}$
C
$-ae^{-a}$
D
$Ae^a$

✓

Answer

Correct option: B.

$-Ae^{-a}$

b
$I = \int\limits_{a - 1}^a {\frac{{{e^{ - t}}}}{{t - a - 1}}\,\,dt} $ put $t = a-1+ y$ (so that lower limit becomes zero)
$\therefore$ $I =$ $\int\limits_0^1 {\frac{{{e^{1 - a - y}}}}{{y - 2}}\,\,\,dy} $ (now using king)
$I =\int\limits_0^1 {\frac{{{e^{1 - a - 1 + y}}}}{{1 - y - 2}}\,\,\,dy} $ $= - {e^{ - a}}\int\limits_0^1 {\frac{{{e^y}}}{{1 + y}}\,\,dy} $ $= - e^{-a} A$

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 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The maximum value of $\mathrm{Z}=x+3 y$ subject to the constraints $2 x+y \leq 20, x+2 y \leq 20$ $x \geq 0, y \geq 0$ is $....$

Linear programming used to optimize mathematical procedure and is:

Subset of mathematical programming
Dimension of mathematical programming
Linear mathematical programming
All of above

The vector which is parallel to the resultant vector of $\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-2 \hat{j}+\hat{k}$ and having magnitude of 5 unit, is __________ .

If $4i+j-k$ and $3i+mj+2k$ are at right angle, then $m = $

${\cot ^{ - 1}}3 + {\rm{cose}}{{\rm{c}}^{ - 1}}\sqrt 5 =$

Let $F: R \rightarrow R$ be a thrice differentiable function. Suppose that $F (1)=0, F (3)=-4$ and $F^{\prime}( x )<0$ for all $x \in$ $(1 / 2,3)$. Let $f(x)=x F(x)$ for all $x \in R$.

$1.$ The correct statement$(s)$ is(are)

$(A)$ $f^{\prime}(1) < 0$

$(B)$ $f(2) < 0$

$(C)$ $f^{\prime}(x) \neq 0$ for any $x \in(1,3)$

$(D)$ $f^{\prime}(x)=0$ for some $x \in(1,3)$

$2.$ If $\int_1^3 x^2 F^{\prime}(x) d x=-12$ and $\int_1^3 x^3 F^{\prime \prime}(x) d x=40$, then the correct expression$(s)$ is(are)

$(A)$ $9 f^{\prime}(3)+f^{\prime}(1)-32=0$

$(B)$ $\int_1^3 f(x) d x=12$

$(C)$ $9 f^{\prime}(3)-f^{\prime}(1)+32=0$

$(D)$ $\int_1^3 f(x) d x=-12$

Give the answer question $1$ and $2.$

If lines $\frac{{x - 1}}{3} = \frac{{y - 2}}{{ - 1}} = \frac{{z - \lambda }}{2}$ and $\frac{{x + 1}}{{ - 2}} = \frac{y}{{3\lambda }} = \frac{{2z - 7}}{1}$ are coplanar then sum of value $(s)$ of $\lambda $ is

Let $v$ be the solution of the differential equation $\left(1-x^{2}\right) d y=\left(xy+\left(x^{3}+2\right) \sqrt{1-x^{2}}\right) d x,-1 < x < 1$ and $y(0)=0$ if $\int\limits_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^{2}} y(x) d x=k$ then $k^{-1}$ is equal to:

For any square matrix $A, A+A^{\prime}$ will be :

$\int_{\,0}^{\,1} {\,\sin \left( {2{{\tan }^{ - 1}}\sqrt {\frac{{1 + x}}{{1 - x}}} } \right)\,dx = } $