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

The integral of ((x+1)^3)/x is (x^3)/3+(3x^2)/2+3x+ln|x|+C

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

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

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

Solution steps

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

Expand

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

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

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

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

\int 3 d x = 3 x

\int \frac{1}{x} d x = ln ∣ x ∣

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

Add a constant to the solution

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

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What is the integral of ((x+1)^3)/x ?
The integral of ((x+1)^3)/x is (x^3)/3+(3x^2)/2+3x+ln|x|+C