Question
$\int\limits^{\frac{\pi}{2}}_0\sin2\text{x }\log\tan\text{x dx}$ is equal to:
-
$\pi$
-
$\frac{\pi}{2}$
-
$0$
-
$2\pi$
$\pi$
$\frac{\pi}{2}$
$0$
$2\pi$
Solution:
$\text{I}=\int\limits^{\frac{\pi}{2}}_0\sin2\text{x }\log\tan\text{x dx}\ ....(\text{i})$
$\text{I}=\int\limits^{\frac{\pi}{2}}_0\sin(\pi-2\text{x})\log\tan\big(\frac{\pi}{2}-\text{x}\big)\text{dx}$
$\text{I}=\int\limits^{\frac{\pi}{2}}_0\sin2\text{x}\log\cot\text{x dx}\ ...(\text{ii})$
Adding (i) and (ii) we get
$2\text{I}=\int\limits^{\frac{\pi}{2}}_0\sin2\text{x}\big(\log\tan\text{x}+\log\cot\text{x}\big)\text{dx}$
$2\text{I}=\int\limits^{\frac{\pi}{2}}_0\sin2\text{x}\big(\log\tan\text{x}\cot\text{x}\big)\text{dx}$
$2\text{I}=\int\limits^{\frac{\pi}{2}}_0\sin\text{x}(\log1)\text{dx}$
$\text{I}=0$
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$STATEMENT -1$ : The probability that the system of equations has a unique solution is $3 / 8$. and $STATEMENT - 2$: The probability that the system of equations has a solution is $1$ .