Question

$\begin{gathered} \rm If \: f(x) = \int \frac{5 {x}^{8} + 7 {x}^{6} }{( {x {}^{2} + 1 + 2 {x}^{7} {)}}^{2} } \: dx \\ \\ \rm (x \geqslant 0) \: and \: f(0) = 0\\ \end{gathered}$

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$\rm Then \: Find \: the \: value \: of \: f(1)$
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Answer 1

ANSWER :-

$\rm \int \frac{5 {x}^{8} + 7 {x}^{6} }{( {x {}^{2} + 1 + 2 {x}^{7} {)}}^{2} } \: \small dx \\ \\ \\ \int \frac{ {5x }^{ - 6 } { \: + \: 7x}^{ - 8} }{ (\frac{1}{ {x}^{7} } + \frac{1}{{x}^{5} } + 2)^{2} } dx = \frac{1}{2 + \frac{1}{ {x}^{5} } + \frac{1}{ {x}^{7} } } + C \\ \\ \\ As \: f \: (0) \: = 0, \: f \: (x) = \frac{ {x}^{7} }{ {2x}^{7} + {x}^{2} + 1} \\ \\ \\ f \: (1) = \frac{1}{4} (Ans)$