MCQ
$\int\limits_{ - 1}^0 {\frac{{4{x^2} + 4x + 3}}{{1 + {e^{2x + 1}}}}} dx\, = $
- A$\frac{7}{3}$
- B$0$
- ✓$\frac{7}{6}$
- D$\frac{7}{12}$
$\text { Put } 2 x+1=t, d x=\frac{d t}{2}$
$I=\frac{1}{2} \int_{-1}^{1} \frac{t^{2}+2}{1+e^{t}}$
$2 \mathrm{I}=\frac{1}{2} \int_{-1}^{1}\left(\mathrm{t}^{2}+2\right) \mathrm{dt}$
$2 \mathrm{I}=\frac{1}{2} \times 2 \int_{0}^{1}\left(\mathrm{t}^{2}+2\right) \mathrm{dt}$
$2 \mathrm{I}=\left(\frac{t^{3}}{3}+2 t\right)_{0}^{1}$
$I=\frac{7}{6}$
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$E_1$ : Six fair dice are rolled and at least one die shows six.
$E_2$ : Twelve fair dice are rolled and at least two dice show six.
Let $p_1$ be the probability of $E_1$ and $p_2$ be the probability of $E_2$. Which of the following is true?