QJEEM19Maf11c: JEE Main 2019 Mathematics - Class 12 JEEM19Maf11c Solution: If        then the value of is :

Question

Verified Solution • 2025–26 Session moderate

If

f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx,\,\left( {x \ge 0} \right),

f\left( 0 \right) = 0,

then the value of

f(1)

is :

Accepted Answer

f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx

f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{x^{14}}{{\left( {{x^{ - 5}} + {x^{ - 7}} + 2} \right)}^2}}}} \,dx

f\left( x \right) = \int {{{5{x^{ - 6}} + 7{x^{ - 8}}} \over {{{\left( {{x^{ - 5}} + {x^{ - 7}} + 2} \right)}^2}}}} \,dx

Let

{x^{ - 5}} + {x^{ - 7}} + 2 = t

\left( { - 5{x^{ - 6}} - 7{x^{ - 8}}} \right)dx = dt

\left( {5{x^{ - 6}} + 7{x^{ - 8}}} \right)dx = - dt

f(x) = \int {{{ - dt} \over {{t^2}}}} = {1 \over t} + c

f\left( x \right) = {1 \over {{x^{ - 5}} + {x^{ - 7}} + 2}} + c

f\left( x \right) = {{{x^7}} \over {{x^2} + 1 + 2{x^7}}} + c

f\left( 0 \right) = 0

\therefore

c = 0

f\left( x \right) = {{{x^7}} \over {\left( {{x^2} + 1 + 2{x^7}} \right)}}

f(1) = {1 \over {1 + 1 + 2}} = {1 \over 4}

QJEEM19Maf11c: JEE Main 2019 Mathematics - Class 12 JEEM19Maf11c Solution: If&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; then the value of is :