QJEEM21Ma67d7: JEE Main 2021 Mathematics - Class 12 JEEM21Ma67d7 Solution: If , where C is a constant of integration, then u + v is equal…

Question

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If

\int {{{2{e^x} + 3{e^{ - x}}} \over {4{e^x} + 7{e^{ - x}}}}dx = {1 \over {14}}(ux + v{{\log }_e}(4{e^x} + 7{e^{ - x}})) + C}

, where C is a constant of integration, then u + v is equal to _____________.

Accepted Answer

2{e^x} + 3{e^{ - x}} = A(4{e^x} + 7{e^{ - x}}) + B(4{e^x} - 7{e^{ - x}}) + \lambda

2 = 4A + 4B ; 3 = 7A

-

7B ;

\lambda

= 0

A + B = {1 \over 2}

A - B = {3 \over 7}

A = {1 \over 2}\left( {{1 \over 2} + {3 \over 7}} \right) = {{7 + 6} \over {28}} = {{13} \over {28}}

B = A - {3 \over 7} = {{13} \over {28}} - {3 \over 7} = {{13 - 12} \over {28}} = {1 \over {28}}

\int {{{13} \over {28}}dx + {1 \over {28}}\int {{{4{e^x} - 7{e^{ - x}}} \over {4{e^x} + 7{e^{ - x}}}}} dx}

=

{{13} \over {28}}x + {1 \over {28}}\ln |4{e^x} + 7{e^{ - x}}| + C

u = {{13} \over 2};v = {1 \over 2}

\Rightarrow

u + v = 7

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