Demonstrate that `x+(1/2)ln(x^2+2)-(sqrt2)arc tan (x/sqrt2)`  is anti-derivative of `(x^2+x)/(x^2+2)`

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hala718 eNotes educator| Certified Educator

We need to determine that:

`x+ (1/2)ln(x^2+2) - (sqrt2)arctan(x/sqrt2)`

is anti-derivative of the function:


`int (x^2+x)/(x^2+2)dx``` = ?

First we will separate into two integrals.

`= int x^2/(x^2+2) dx + int x/(x^2+2) dx`

Now we will use the integration tables to determine the integral of both functions.

We know that:

`int x^2/(a^2+x^2) dx = x-a*arctann(x/a)`

`==> int x^2/(x^2+2) dx = x- (sqrt2)arctan(x/sqrt2)`

`int x/(x^2+a^2) dx = (1/2)ln(a^2+x^2)`

`==> int x/(x^2+2) dx = (1/2)ln(x^2+2)`

`==> int (x^2+x)/(x^2+2) dx = x+ (1/2)ln(x^2+2) - (sqrt2)arctan(x/sqrt2)`