Question

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