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

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

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

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

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

Solution steps

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

Long division

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

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

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

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

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

Add a constant to the solution

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

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