Question # If ( int frac{1}{x+x^{5}} d x=f(x)+c, ) then ( int frac{x^{4}}{x+x^{5}} d x= )

# If ( int frac{1}{x+x^{5}} d x=f(x)+c, ) then ( int frac{x^{4}}{x+x^{5}} d x= )

1) ( log |x|+f(x)+c_{1} )

2) ( log |x|-f(x)+k )

3) ( x f(x)+c_{1} )

4) ( x log |x|+c_{1} )

Solution

( begin{aligned} & int frac{x^{4}}{x+x^{5}} d x=int frac{left(x^{4}+2right)-t}{xleft(1+x^{4}right)} d x &=int frac{d x}{x}-int frac{d x}{xleft(1+x^{4}right)} &=log |x|-f(x)+k end{aligned} )