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

The integral of (x^3+1)/x is (x^3)/3+ln|x|+C

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integral of (x^3+1)/x

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

\frac{x ^{3}}{3} + ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{x ^{3}}{3} + ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{3}}{3} + ln ∣ x ∣ + C

y^{^{'}} = 2 y^{2}

y (1 + x^{2}) y^{^{'}} - x (1 + y^{2}) = 0, y (0) = 3

\int \frac{3 ln ^{2} ( x ) + 36}{x} d x

x \to - 8 lim (12 x + 11)

\frac{d}{d x} (\frac{1}{x} - \frac{4}{x ^{2}})

What is the integral of (x^3+1)/x ?
The integral of (x^3+1)/x is (x^3)/3+ln|x|+C