Question

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

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

Answer 1

The correct option is D

$\frac{1}{4}$

$f\left(x\right)=\int \frac{5x^8+7x^6}{x^{14}{(2+\frac{1}{x^7}+\frac{1}{x^5})}^2}dx$
$=\int \frac{\frac{5}{x^6}+\frac{7}{x^8}}{{(2+\frac{1}{x^7}+\frac{1}{x^5})}^2}dx$
Put, $2+\frac{1}{x^7}+\frac{1}{x^5}=t$
dt= -dx($\frac{5}{x^6}+\frac{7}{x^8}$)
$\Rightarrow f\left(x\right)=\frac{-dt}{t^2}=\frac{1}{t}+c$
$=\left(\frac{x^7}{2x^7+x^2+1}\right)+c$
$f\left(0\right)=0\Rightarrow c=0$
$\Rightarrow f\left(x\right)=-\left(\frac{x^7}{2x^7+x^2+1}\right)\Rightarrow f\left(1\right)=\frac{1}{4}$