Question

If $f\left(x\right) = \int \frac{5x^{8} +7x^{6}}{\left(x^{2}+1+2x^{7}\right)^{2}} dx , \left(x \ge0\right) $ and $f(0) = 0$, then the value of $f(1)$ is :

• A. $ - \frac{1}{2}$
• B. $ \frac{1}{2}$
• C. $ - \frac{1}{4}$
• D. $ \frac{1}{4}$

Answer 1

Answer: (D)

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