Mathematics

$$\displaystyle \int_{0}^{1}\frac{2^{x+1}-3^{x-1}}{6^{x}}dx$$


ANSWER

$$\displaystyle \frac{4}{3}\log _{3}e-\frac{1}{6}\log _{2}e.$$


SOLUTION
$$ \displaystyle \int _{ 0 }^{ 1 } \frac { 2^{ x+1 }-3^{ x-1 } }{ 6^{ x } } dx=\int _{ 0 }^{ 1 } \left( 2.3^{ -x }-\frac { 1 }{ 3 } 2^{ -x } \right) dx$$

$$ \displaystyle =\left[ -2\frac { 3^{ -x } }{ \log  3 } +\frac { 1 }{ 3 } \frac { 2^{ -x } }{ \log  2 }  \right] ^{ 1 }$$

$$ \displaystyle =-\frac { 2 }{ \log  3 } \left( \frac { 1 }{ 3 } -1 \right) +\frac { 1 }{ 3\log  2 } \left( \frac { 1 }{ 2 } -1 \right) $$

$$\displaystyle =\frac { 4 }{ 3 } \log _{ 3 } e-\frac { 1 }{ 6 } \log _{ 2 } e$$
Hence, option 'A' is correct.
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Single Correct Medium Published on 17th 09, 2020
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