The value of definite integral _ ^0 z , e^ - z 1 - e^ - 2z ,dz . - Vidyadip

MCQ

The value of definite integral $\int\limits_\infty ^0 {\frac{{z\,{e^{ - z}}}}{{\sqrt {1 - {e^{ - 2z}}} }}\,dz} $.

✓
$-\frac{\pi }{2}\,\ln 2$
B
$\frac{\pi }{2}\,\ln 2$
C
$- \pi\, ln\, 2$
D
$\pi \,ln\, 2$

✓

Answer

Correct option: A.

$-\frac{\pi }{2}\,\ln 2$

a
$l=\int_{\infty}^{0} \frac{z e^{-z}}{\sqrt{1-e^{-2 z}}} d z$ put $e^{-z}=\sin \theta$

$l=-\int_{0}^{\pi / 2} \frac{\ln (\sin \theta)(-\cos \theta) d \theta}{\sqrt{1-\sin ^{2} \theta}}=\int_{0}^{\pi / 2} \ln \sin \theta d \theta$

$\frac{-\pi}{2} \ln 2$

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 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

What are the direction cosines of a line which is equally inclined to the positive directions of the axes:

$\Big(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\Big)$
$\Big(-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\Big)$
$\Big(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\Big)$
$\Big(\frac{1}{3},\frac{1}{3},\frac{1}{3}\Big)$

If $\mathrm{p}(\mathrm{x})$ be a polynomial of degree three that has a local maximum value $8$ at $x=1$ and a local minimum value $4$ at $x=2 ;$ then $p(0)$ is equal to

$\int_{ - \pi /4}^{\pi /2} {{e^{ - x}}\sin x\,dx} = $

The vector equation of the line $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$ is

$\int|\text{x}|^3\text{ dx}$ is equal to:

$\frac{-\text{x}^4}{4}+\text{C}$
$\frac{|\text{x}|^4}{4}+\text{C}$
$\frac{\text{x}^4}{4}+\text{C}$
none of these.

If $f(x) = 3{e^{{x^2}}}$, then $f'(x) - 2xf(x) + {1 \over 3}f(0) - f'(0) = $

If $4{\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \pi ,$ then $x$ is equal to

A bag contains $a$ white and $b$ black balls. Two players $A$ and $B$ alternately draw a ball from the bag replacing the ball each time after the draw till one of them draws a white ball and wins the game. $A$ begins the game. If the probability of $A$ winning the game is three times that of $B$, then the ratio $a : b$ is

If $A$, $B$ and $C$ are square matrices of order $3$ such that $A = \left[ {\begin{array}{*{20}{c}} x&0&1 \\ 0&y&0 \\ 0&0&z \end{array}} \right]$ and $\left| B \right| = 36$, $\left| C \right| = 4$, $\left( {x,y,z \in N} \right)$ and $\left| {ABC} \right| = 1152$ then the minimum value of $x + y + z$ is

A constraint in an LP model becomes redundant because: