MCQ
$\int\limits_0^{\ln \,\,5} {\,\frac{{{e^x}\,\,\sqrt {{e^x}\, - \,1} }}{{{e^x}\, + \,3}}} $ $dx =$
- ✓$4 - \pi$
- B$6 - \pi$
- C$5 - \pi$
- D$None$
Substitute $t=e^{x} \Rightarrow d t=e^{x} d x,$ we get
$I=\int \frac{\sqrt{t-1}}{t+3} d t$
Substitute $u=\sqrt{t-1} \Rightarrow d u=\frac{1}{2 \sqrt{t-1}} d t,$ we get
$I=2 \int \frac{u^{2}}{u^{2}+4} d u=2 \int\left(1-\frac{4}{u^{2}+4}\right) d u=2 \int d u-2 \int \frac{4}{u^{2}+4} d u$
$=2 u-4 \tan ^{-1}\left(\frac{u}{2}\right)=2 \sqrt{t-1}-4 \tan ^{-1}\left(\frac{\sqrt{t-1}}{2}\right)$
$=2 \sqrt{e^{x}-1}-4 \tan ^{-1}\left(\frac{\sqrt{e^{x}-1}}{2}\right)$
Therefore, $\int_{0}^{\log 5} I d x=\left[2 \sqrt{e^{x}-1}-4 \tan ^{-1}\left(\frac{\sqrt{e^{x}-1}}{2}\right)\right]_{0}^{\log 5}=4-\pi$
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Statement $I$ : For any two non-zero complex numbers $\mathrm{z}_1, \mathrm{z}_2$
$\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)$ and
Statement $II$ : If $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are three distinct complex numbers and a, b, c are three positive real numbers such that $\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}$, then
$\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1$
Between the above two statements,