What is the integral of (2x^3-1)/(x^3) ?

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

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integral of (2x^3-1)/(x^3)

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

2 x + \frac{1}{2 x ^{2}} + C

Solution steps

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

Expand

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

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

\int 2 d x = 2 x

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

= 2 x - (- \frac{1}{2 x ^{2}})

Simplify

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

Add a constant to the solution

= 2 x + \frac{1}{2 x ^{2}} + C

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