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

The integral of (5+x^3+2x)/x is 5ln|x|+(x^3)/3+2x+C

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

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

5 ln ∣ x ∣ + \frac{x ^{3}}{3} + 2 x + C

Solution steps

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

Expand

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

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

\int 2 d x = 2 x

= 5 ln ∣ x ∣ + \frac{x ^{3}}{3} + 2 x

Add a constant to the solution

= 5 ln ∣ x ∣ + \frac{x ^{3}}{3} + 2 x + C

\frac{\partial}{\partial x} (3 (x - 6)^{3})

3 - 11 x^{5}

\frac{d}{d x} (\frac{x ^{3} + 1}{x ^{2} - 1})

n = 1 \sum \infty \frac{n}{8 ^{n}}

\frac{\partial}{\partial y} (y sin (x z))

What is the integral of (5+x^3+2x)/x ?
The integral of (5+x^3+2x)/x is 5ln|x|+(x^3)/3+2x+C