Question

x⁶+2/x²+1,Integrate the given function w.r.t. x considering them well defined and integrable over proper domain.

Answer 1

X raise power 6+1 divide 6+1 than+ 2x raise power -2+1 divide by -2+1than+ x

Answer 2

we have to find the value of $\int{\frac{x^6+2}{x^2+1}}\,dx$

x^6 + 2 = (x²)³ + 1³ + 1
use formula, a³ + b³ = (a + b)(a² - ab + b³)
= (x² + 1)(x⁴ - x² + 1) + 1

so, $\int{\frac{x^6+2}{x^2+1}}\,dx=\int{\frac{(x^2+1)(x^4-x^2+1)+1}{x^2+1}}\,dx$

$=\int{(x^4-x^2+1)+\frac{1}{x^2+1}}\,dx$

$=\int{x^4}\,dx-\int{x^2}\,dx+\int{dx}+\int{\frac{1}{x^2+1}}\,dx$

$=\frac{x^5}{5}-\frac{x^3}{3}+x+tan^{-1}x+C$