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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Simplify

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

Add a constant to the solution

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

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