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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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