What is the integral of ((x^6+5x+3)/x) ?

The integral of ((x^6+5x+3)/x) is (x^6)/6+5x+3ln|x|+C

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integral of ((x^6+5x+3)/x)

\int (\frac{x ^{6} + 5 x + 3}{x}) d x

\frac{x ^{6}}{6} + 5 x + 3 ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{6} + 5 x + 3}{x} d x

Expand

\frac{x ^{6} + 5 x + 3}{x} : x^{5} + 5 + \frac{3}{x}

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

\int x^{5} d x = \frac{x ^{6}}{6}

\int 5 d x = 5 x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

= \frac{x ^{6}}{6} + 5 x + 3 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{6}}{6} + 5 x + 3 ln ∣ x ∣ + C

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What is the integral of ((x^6+5x+3)/x) ?
The integral of ((x^6+5x+3)/x) is (x^6)/6+5x+3ln|x|+C