MCQ
$\int \limits_0^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+11 e^x+6} d x$
- A$\log _e\left(\frac{512}{81}\right)$
- ✓$\log _e\left(\frac{32}{27}\right)$
- C$\log _e\left(\frac{256}{81}\right)$
- D$\log _e\left(\frac{64}{27}\right)$
$=6 \int \limits_0^{\infty}\left(\frac{\frac{1}{2}}{ e ^{ x }+1}+\frac{-1}{ e ^{ x }+2}+\frac{\frac{1}{2}}{ e ^{ x }+3}\right) d x$
$=3 \int \limits_0^{\infty} \frac{ e ^{- x }}{1+ e ^{- x }} dx -6 \int \limits_0^{\infty} \frac{ e ^{- x } dx }{1+2 e ^{- x }}+3 \int \limits_0^{\infty} \frac{ e ^{- x }}{1+3 e ^{- x }} dx$
$=3\left[-\ln \left(1+ e ^{- x }\right)\right]_0^{\infty}+6 \frac{1}{2}\left[\ln \left(1+2 e ^{- x }\right)\right]_0^{\infty}$
$-\frac{3}{3}\left[\ln \left(1+3 e ^{- x }\right)\right]_0^{\infty}$
$=3 \ln 2-3 \ln 3+\ln 4$
$=3 \ln \frac{2}{3}+\ln 4$
$=\ln \frac{32}{27}$
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$\left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right] x \frac{d y}{d x}=x+\left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right] y$
pass through the points $(1,0)$ and $(2 \alpha, \alpha), \alpha>0$.
Then $\alpha$ is equal to