Question #0345c

1 Answer

If #color(red)(int ((x^2+x+1)dx)/(x^2+1)=x+1/2ln(x^2+1)+C)#

If #color(red)(int (x^2+x+1/x^2+1)dx=x^3/3+x^2/2-1/x+x+C)#

Explanation:

I can see two possibilities in the problem that you wrote.

If the problem is

#color(blue)(int ((x^2+x+1)dx)/(x^2+1))#

The solution is

#int ((x^2+x+1)dx)/(x^2+1)=int ((x^2+1+x)dx)/(x^2+1)=int ((x^2+1)/(x^2+1)+x/(x^2+1))dx=int (1+x/(x^2+1))dx#

#int ((x^2+x+1)dx)/(x^2+1)=int dx+1/2int ((2x)/(x^2+1))dx#

#color(red)(int ((x^2+x+1)dx)/(x^2+1)=x+1/2ln(x^2+1)+C)#

If the problem is

#color(blue)(int (x^2+x+1/x^2+1)dx)#

#color(red)(int (x^2+x+1/x^2+1)dx=x^3/3+x^2/2-1/x+x+C)#

God bless....I hope the explanation is useful.