What is the integral of ((x^3+1)/(x^5)) ?

The integral of ((x^3+1)/(x^5)) is -1/x-1/(4x^4)+C

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integral of ((x^3+1)/(x^5))

\int (\frac{x ^{3} + 1}{x ^{5}}) d x

- \frac{1}{x} - \frac{1}{4 x ^{4}} + C

Solution steps

\int \frac{x ^{3} + 1}{x ^{5}} d x

Expand

\frac{x ^{3} + 1}{x ^{5}} : \frac{1}{x ^{2}} + \frac{1}{x ^{5}}

= \int \frac{1}{x ^{2}} + \frac{1}{x ^{5}} 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{1}{x ^{2}} d x + \int \frac{1}{x ^{5}} d x

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

\int \frac{1}{x ^{5}} d x = - \frac{1}{4 x ^{4}}

= - \frac{1}{x} - \frac{1}{4 x ^{4}}

Add a constant to the solution

= - \frac{1}{x} - \frac{1}{4 x ^{4}} + C

What is the integral of ((x^3+1)/(x^5)) ?
The integral of ((x^3+1)/(x^5)) is -1/x-1/(4x^4)+C