What is the integral of (4/x+2x^3-2/(x^3)) ?

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

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integral of (4/x+2x^3-2/(x^3))

\int (\frac{4}{x} + 2 x^{3} - \frac{2}{x ^{3}}) d x

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

Solution steps

\int \frac{4}{x} + 2 x^{3} - \frac{2}{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 \frac{4}{x} d x + \int 2 x^{3} d x - \int \frac{2}{x ^{3}} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int 2 x^{3} d x = \frac{x ^{4}}{2}

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

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

Simplify

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

Add a constant to the solution

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

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