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 $e^{-z}=\sin \theta$

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

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

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