Question

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

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

Answer 1

Answer: (D)

$\frac{1}{4}$

$\int \frac{5x^8+7x^6}{(x^2+1+2x^7{)}^2}dx$

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

As $f\left(0\right)=0,f\left(x\right)=\frac{x^7}{2x^7+x^2+1}$

$f\left(1\right)=\frac{1}{4}$