If f(x)=∫5×8+7x6x2+1+2x72dx,(x≥0), and f(0)=0, then the value of F(1) is

If $f (x) = \int \frac{5 x^{8} + 7 x^{6}}{{(x^{2} + 1 + 2 x^{7})}^{2}} d x, (x \geq 0)$ , and $f (0) = 0$ , then the value of $F (1)$ is

A
B
C
D

Solution:

We have, $f (x) = \int \frac{5 x^{8} + 7 x^{6}}{{(x^{2} + 1 + 2 x^{7})}^{2}} d x$

$= \int \frac{5 x^{- 6} + 7 x^{- 8}}{{(\frac{1}{x^{5}} + \frac{1}{x^{7}} + 2)}^{2}} d x = \frac{1}{2 + \frac{1}{x^{5}} + \frac{1}{x^{7}}} + C$

$L e t, 2 + \frac{1}{x^{5}} + \frac{1}{x^{7}} = t$

Since, $f (0) = 0 \Rightarrow C = 0$

$∴ f (x) = \frac{x^{7}}{2 x^{7} + x^{2} + 1}$ . Now, $F (1) = \frac{1}{4}$

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