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Answer :
On evaluating the definite integral [tex]\(\int_{1}^{2} \frac{2}{x} \,dx\)[/tex] , we get D) [tex]\[ 2 \ln(2)\][/tex].
1) To evaluate the definite integral [tex]\(\int_{1}^{2} \frac{2}{x} \,dx\)[/tex], we integrate the function 2/x with respect to x over the interval [1, 2].
[tex]\[\int_{1}^{2} \frac{2}{x} \,dx = 2 \int_{1}^{2} \frac{1}{x} \,dx\][/tex]
2) Now, let's integrate [tex]\(\frac{1}{x}\)[/tex] with respect to x:
[tex]\[= 2 \left[ \ln|x| \right]_{1}^{2}\][/tex]
[tex]\[= 2 \left( \ln|2| - \ln|1| \right)\][/tex]
[tex]\[= 2 \ln|2| - 2 \ln|1|\][/tex]
[tex]\[= 2 \ln|2| - 2 \cdot 0\][/tex]
[tex]\[= 2 \ln|2|\][/tex]
[tex]\[= \ln(2^2)\][/tex]
[tex]\[= \ln(4)\][/tex]
So, the correct answer is [tex]\[D) \, 2 \ln(2)\][/tex]
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