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

The integral of (x+1/x)^3 is (x^4)/4+(3x^2)/2+3ln|x|-1/(2x^2)+C

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integral of (x+1/x)^3

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

\frac{x ^{4}}{4} + \frac{3 x ^{2}}{2} + 3 ln ∣ x ∣ - \frac{1}{2 x ^{2}} + C

Solution steps

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

Expand

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

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

\int x^{3} d x = \frac{x ^{4}}{4}

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

= \frac{x ^{4}}{4} + \frac{3 x ^{2}}{2} + 3 ln ∣ x ∣ - \frac{1}{2 x ^{2}}

Add a constant to the solution

= \frac{x ^{4}}{4} + \frac{3 x ^{2}}{2} + 3 ln ∣ x ∣ - \frac{1}{2 x ^{2}} + C

What is the integral of (x+1/x)^3 ?
The integral of (x+1/x)^3 is (x^4)/4+(3x^2)/2+3ln|x|-1/(2x^2)+C