QJEEM21Ma6696: JEE Main 2021 Mathematics - Class 12 JEEM21Ma6696 Solution: If and , then the value of K is

Question

Verified Solution • 2025–26 Session moderate

If

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

and

f(1) = {1 \over K}

, then the value of K is

Accepted Answer

\int {{{5{x^8} + 7{x^6}} \over {{{(2{x^7} + {x^2} + 1)}^2}}}dx = \int {{{5{x^8} + 7{x^6}} \over {{x^{14}}{{\left( {2 + {1 \over {{x^5}}} + {1 \over {{x^7}}}} \right)}^2}}}dx} }

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

put

2 + {1 \over {{x^5}}} + {1 \over {{x^7}}} = t

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

\int {{{ - dt} \over {{t^2}}} = {1 \over t} + c}

\Rightarrow f(x) = {1 \over {2 + {1 \over {{x^5}}} + {1 \over {{x^7}}}}} + C = {{{x^7}} \over {2{x^7} + 1 + {x^2}}} + C

f(0) = 0 \Rightarrow C = 0

f(x) = {1 \over 4} = {1 \over k}

\Rightarrow k = 4

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