• # question_answer The value of the integral $\int\limits_{0}^{2}{\frac{\log \,({{x}^{2}}+2)}{{{(x+2)}^{2}}}\,dx}$ is A)  $\frac{\sqrt{2}}{3}{{\tan }^{-1}}\sqrt{2}+\frac{5}{12}\,\log \,2-\frac{1}{4}\,\log \,3$ B)  $\frac{\sqrt{2}}{3}{{\tan }^{-1}}\sqrt{2}-\frac{5}{12}\,\log \,2-\frac{1}{12}\,\log \,3$ C)  $\frac{\sqrt{2}}{3}{{\tan }^{-1}}\sqrt{2}+\frac{5}{12}\,\log \,2+\frac{1}{4}\,\log \,3$ D) $\frac{\sqrt{2}}{3}{{\tan }^{-1}}\sqrt{2}-\frac{5}{12}\,\log \,2+\frac{1}{\sqrt{12}}\,\log \,3$

Solution :

Let $u=v,$ $u=v+at$ $0=v-at$ $\therefore$ $-a=\frac{0-v}{t}=-\frac{v}{t}$ $f=\mu R=\mu mg$ (resolved in partially) $a=\mu g$ $\therefore$ $t=\frac{v}{a}=\frac{v}{g\mu }$ $\eta =\frac{{{P}_{0}}}{{{P}_{i}}}$ $\therefore$ $\frac{1}{2}\,m{{v}^{2}}=16\,\,J$ $v=4\,m{{s}^{-1}}$ $Mg\,\,\sin \,\,\theta$ $Mg\,\,\sin \theta \times \frac{h}{2}Mg\,\cos \,\theta \,\times r$

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