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

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

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

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

\frac{x ^{3}}{3} + 2 x + arctan (x) + C

Solution steps

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

Expand

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

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

\int \frac{x ^{4}}{x ^{2} + 1} d x = \frac{x ^{3}}{3} - x + arctan (x)

\int \frac{3 x ^{2}}{x ^{2} + 1} d x = 3 (- arctan (x) + x)

\int \frac{3}{x ^{2} + 1} d x = 3 arctan (x)

= \frac{x ^{3}}{3} - x + arctan (x) + 3 (- arctan (x) + x) + 3 arctan (x)

Simplify

\frac{x ^{3}}{3} - x + arctan (x) + 3 (- arctan (x) + x) + 3 arctan (x) : \frac{x ^{3}}{3} + 2 x + arctan (x)

= \frac{x ^{3}}{3} + 2 x + arctan (x)

Add a constant to the solution

= \frac{x ^{3}}{3} + 2 x + arctan (x) + C

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