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

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

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

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

\frac{2 x ^{3}}{3} - x + 4 arctan (x) + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

2 (\frac{x ^{3}}{3} - x + arctan (x)) - arctan (x) + x + 3 arctan (x) : \frac{2 x ^{3}}{3} - x + 4 arctan (x)

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

Add a constant to the solution

= \frac{2 x ^{3}}{3} - x + 4 arctan (x) + C

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