QJEEM21Ma7f08: JEE Main 2021 Mathematics - Class 12 JEEM21Ma7f08 Solution: The integral , x > 0, is equal to : (where c is a constant of…

Question

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The integral

\int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} - 7{e^{2{{\log }_e}x}}}}} dx

, x > 0, is equal to : (where c is a constant of integration)

Accepted Answer

\int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} - 7{e^{2{{\log }_e}x}}}}dx}

= \int {{{8{x^3} + 5(4{x^2})} \over {{x^4} + 5{x^3} - 7{x^2}}}}

= \int {{{8{x^3} + 20{x^2}} \over {{x^4} + 5{x^3} - 7{x^2}}}}

= \int {{{8x + 20} \over {{x^2} + 5x - 7}}}

= \int {{{4(2x + 5)} \over {{x^2} + 5x - 7}}}

Let

{\begin{matrix} x^{2} + 5 x - 7 = t \\ (2 x + 5) d x = d t \end{matrix}}

= \int {{{4dt} \over t}}

= 4\ln \left| t \right| + C

= 4\ln \left| {({x^2} + 5x - 7)} \right| + C

Here C is integral constant.

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