Question

If $f\left(x\right)=\int \frac{x^8+4}{x^4-2x^2+2}dx$ and $f\left(0\right)=0$ , then

• A. $f\left(x\right)$ is an odd function
• B. $f\left(x\right)$ has range $R$
• C. $f\left(x\right)$ has at least one real root
• D. $f\left(x\right)$ is a monotonic function

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $f\left(x\right)$ is an odd function
B $f\left(x\right)$ has range $R$
C $f\left(x\right)$ has at least one real root
D $f\left(x\right)$ is a monotonic function

$f\left(x\right)=\int \frac{x^8+4}{x^4-2x^2+2}dx$
$\int \frac{(x^8+4+4x^4)-4x^4}{x^4-2x^2+2}dx$
$\int \frac{(x^4+2{)}^2-(2x^2{)}^2}{(x^4-2x^2+2)}dx$
$\int \frac{(x^4+2-2x^2)(x^4+2+2x^2)}{(x^4-2x^2+2)}dx$
$f\left(x\right)=\frac{x^5}{5}+\frac{2x^3}{3}+2x+C$
Since, $f\left(0\right)=0$
$\Rightarrow C=0$
Hence, $f\left(x\right)=\frac{x^5}{5}+\frac{2x^3}{3}+2x$
Clearly, $f\left(x\right)$ is odd function.
Range of $f$ is $R$