Question

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

• A. -1/2
• B. 1/4
• C. 1/2
• D. -1/4

Answer 1

Answer: (B)

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