What is the integral of (x^3+5x+1)/(x^2+5) ?

The integral of (x^3+5x+1)/(x^2+5) is 1/2 x^2+5/2+1/(sqrt(5))arctan(x/(sqrt(5)))+C

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integral of (x^3+5x+1)/(x^2+5)

\int \frac{x ^{3} + 5 x + 1}{x ^{2} + 5} d x

\frac{1}{2} x^{2} + \frac{5}{2} + \frac{1}{5} arctan (\frac{x}{5}) + C

Solution steps

\int \frac{x ^{3} + 5 x + 1}{x ^{2} + 5} d x

Expand

\frac{x ^{3} + 5 x + 1}{x ^{2} + 5} : \frac{x ^{3}}{x ^{2} + 5} + \frac{5 x}{x ^{2} + 5} + \frac{1}{x ^{2} + 5}

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} + 5} d x + \int \frac{5 x}{x ^{2} + 5} d x + \int \frac{1}{x ^{2} + 5} d x

\int \frac{x ^{3}}{x ^{2} + 5} d x = \frac{1}{2} (x^{2} + 5 - 5 ln x^{2} + 5)

\int \frac{5 x}{x ^{2} + 5} d x = \frac{5}{2} ln x^{2} + 5

\int \frac{1}{x ^{2} + 5} d x = \frac{1}{5} arctan (\frac{x}{5})

= \frac{1}{2} (x^{2} + 5 - 5 ln x^{2} + 5) + \frac{5}{2} ln x^{2} + 5 + \frac{1}{5} arctan (\frac{x}{5})

Simplify

\frac{1}{2} (x^{2} + 5 - 5 ln x^{2} + 5) + \frac{5}{2} ln x^{2} + 5 + \frac{1}{5} arctan (\frac{x}{5}) : \frac{1}{2} x^{2} + \frac{5}{2} + \frac{1}{5} arctan (\frac{x}{5})

= \frac{1}{2} x^{2} + \frac{5}{2} + \frac{1}{5} arctan (\frac{x}{5})

Add a constant to the solution

= \frac{1}{2} x^{2} + \frac{5}{2} + \frac{1}{5} arctan (\frac{x}{5}) + C

\frac{\partial}{\partial x} (x e^{\frac{y}{x}})

\int (x \cdot e^{2 \cdot x}) d x

\frac{\partial}{\partial y} (y - ln (x^{2} + y^{2}))

y = \frac{3}{x ^{2} - 1}

x \to 0 + lim (6 (cot (x) - \frac{1}{x}))

What is the integral of (x^3+5x+1)/(x^2+5) ?
The integral of (x^3+5x+1)/(x^2+5) is 1/2 x^2+5/2+1/(sqrt(5))arctan(x/(sqrt(5)))+C