Question

If $\int \frac{x^4+1}{x^6+1}dx={\tan }^{-1}\left(f\right(x\left)\right)-\frac{2}{3}{\tan }^{-1}\left(g\right(x\left)\right)+C$, then

• A. Both $f\left(x\right)$ & $g\left(x\right)$ are odd functions
• B. $g\left(x\right)$ is monotonic function
• C. none of these
• D. None

Answer 1

The correct option is A Both $f\left(x\right)$ & $g\left(x\right)$ are odd functions

$\int \frac{x^4+1}{x^6+1}dx$

$\int \frac{x^4+1}{x^6+1}\times \frac{x^2+1}{x^2+1}dx$

$\int \frac{(x^6+1)+x^2(x^2+1)}{(x^6+1)(x^2+1)}$

$\int (\frac{1}{x^2+1}+\frac{1}{3}\frac{3x^2}{x^6+1})dx$

${\tan }^{-1}x+\frac{1}{3}{\tan }^{-1}\left(x^3\right)+C$