What is the integral of ((x^2+x+1))/((x^2+1)^2) ?

The integral of ((x^2+x+1))/((x^2+1)^2) is arctan(x)-1/(2(x^2+1))+C

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integral of ((x^2+x+1))/((x^2+1)^2)

\int \frac{( x ^{2} + x + 1 )}{( x ^{2} + 1 ) ^{2}} d x

arctan (x) - \frac{1}{2 ( x ^{2} + 1 )} + C

Solution steps

\int \frac{x ^{2} + x + 1}{( x ^{2} + 1 ) ^{2}} d x

Take the partial fraction of

\frac{x ^{2} + x + 1}{( x ^{2} + 1 ) ^{2}} : \frac{1}{x ^{2} + 1} + \frac{x}{( x ^{2} + 1 ) ^{2}}

= \int \frac{1}{x ^{2} + 1} + \frac{x}{( x ^{2} + 1 ) ^{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x ^{2} + 1} d x + \int \frac{x}{( x ^{2} + 1 ) ^{2}} d x

\int \frac{1}{x ^{2} + 1} d x = arctan (x)

\int \frac{x}{( x ^{2} + 1 ) ^{2}} d x = - \frac{1}{2 ( x ^{2} + 1 )}

= arctan (x) - \frac{1}{2 ( x ^{2} + 1 )}

Add a constant to the solution

= arctan (x) - \frac{1}{2 ( x ^{2} + 1 )} + C

x \to - 1 lim (x^{3} - 2 x^{2} + 3 x - 1)

\frac{d}{d x} (25 in (x^{2}) - x^{2})

\int_{a}^{x} \frac{1}{b - a} d x

\frac{x ^{2} + 1}{∣ x ∣ - 1}

\frac{d}{d x} (\frac{cos ( 2 x ) + x ^{3}}{tan ( 2 x ) - 3 x})

What is the integral of ((x^2+x+1))/((x^2+1)^2) ?
The integral of ((x^2+x+1))/((x^2+1)^2) is arctan(x)-1/(2(x^2+1))+C