What is the integral of (x^3+4x-13)/(x^2+4) ?

The integral of (x^3+4x-13)/(x^2+4) is 1/2 x^2+2-13/2 arctan(x/2)+C

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integral of (x^3+4x-13)/(x^2+4)

\int \frac{x ^{3} + 4 x - 13}{x ^{2} + 4} d x

\frac{1}{2} x^{2} + 2 - \frac{13}{2} arctan (\frac{x}{2}) + C

Solution steps

\int \frac{x ^{3} + 4 x - 13}{x ^{2} + 4} d x

Expand

\frac{x ^{3} + 4 x - 13}{x ^{2} + 4} : \frac{x ^{3}}{x ^{2} + 4} + \frac{4 x}{x ^{2} + 4} - \frac{13}{x ^{2} + 4}

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 ^{3}}{x ^{2} + 4} d x + \int \frac{4 x}{x ^{2} + 4} d x - \int \frac{13}{x ^{2} + 4} d x

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

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

\int \frac{13}{x ^{2} + 4} d x = \frac{13}{2} arctan (\frac{x}{2})

= \frac{1}{2} (x^{2} + 4 - 4 ln x^{2} + 4) + 2 ln x^{2} + 4 - \frac{13}{2} arctan (\frac{x}{2})

Simplify

\frac{1}{2} (x^{2} + 4 - 4 ln x^{2} + 4) + 2 ln x^{2} + 4 - \frac{13}{2} arctan (\frac{x}{2}) : \frac{1}{2} x^{2} + 2 - \frac{13}{2} arctan (\frac{x}{2})

= \frac{1}{2} x^{2} + 2 - \frac{13}{2} arctan (\frac{x}{2})

Add a constant to the solution

= \frac{1}{2} x^{2} + 2 - \frac{13}{2} arctan (\frac{x}{2}) + C

2 sin^{5} (x)

\int 13 d x

x \to 7 lim (5 x^{- 1})

\int \frac{x}{x ^{2} - 2 x - 3} d x

x \to 1 lim (\frac{x ^{4} - 1}{x ^{3} - 1})

What is the integral of (x^3+4x-13)/(x^2+4) ?
The integral of (x^3+4x-13)/(x^2+4) is 1/2 x^2+2-13/2 arctan(x/2)+C