What is the integral of (2x^3-x^2+2x+4)/(x^2+1) ?

The integral of (2x^3-x^2+2x+4)/(x^2+1) is x^2-x+5arctan(x)+C

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integral of (2x^3-x^2+2x+4)/(x^2+1)

\int \frac{2 x ^{3} - x ^{2} + 2 x + 4}{x ^{2} + 1} d x

x^{2} - x + 5 arctan (x) + C

Solution steps

\int \frac{2 x ^{3} - x ^{2} + 2 x + 4}{x ^{2} + 1} d x

Long division

\frac{2 x ^{3} - x ^{2} + 2 x + 4}{x ^{2} + 1} : 2 x - 1 + \frac{5}{x ^{2} + 1}

= \int 2 x - 1 + \frac{5}{x ^{2} + 1} 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 2 x d x - \int 1 d x + \int \frac{5}{x ^{2} + 1} d x

\int 2 x d x = x^{2}

\int 1 d x = x

\int \frac{5}{x ^{2} + 1} d x = 5 arctan (x)

= x^{2} - x + 5 arctan (x)

Add a constant to the solution

= x^{2} - x + 5 arctan (x) + C

What is the integral of (2x^3-x^2+2x+4)/(x^2+1) ?
The integral of (2x^3-x^2+2x+4)/(x^2+1) is x^2-x+5arctan(x)+C