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

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

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integral of (1+1/x)^3

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

x + 3 ln ∣ x ∣ - \frac{3}{x} - \frac{1}{2 x ^{2}} + C

Solution steps

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

Expand

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

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

\int 1 d x = x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

= x + 3 ln ∣ x ∣ - \frac{3}{x} - \frac{1}{2 x ^{2}}

Add a constant to the solution

= x + 3 ln ∣ x ∣ - \frac{3}{x} - \frac{1}{2 x ^{2}} + C

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