Question

Question #0345c

Answer 1

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.