What is the integral of (x-4)/(x^9) ?

The integral of (x-4)/(x^9) is -1/(7x^7)+1/(2x^8)+C

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integral of (x-4)/(x^9)

\int \frac{x - 4}{x ^{9}} d x

- \frac{1}{7 x ^{7}} + \frac{1}{2 x ^{8}} + C

Solution steps

\int \frac{x - 4}{x ^{9}} d x

Expand

\frac{x - 4}{x ^{9}} : \frac{1}{x ^{8}} - \frac{4}{x ^{9}}

= \int \frac{1}{x ^{8}} - \frac{4}{x ^{9}} 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 ^{8}} d x - \int \frac{4}{x ^{9}} d x

\int \frac{1}{x ^{8}} d x = - \frac{1}{7 x ^{7}}

\int \frac{4}{x ^{9}} d x = - \frac{1}{2 x ^{8}}

= - \frac{1}{7 x ^{7}} - (- \frac{1}{2 x ^{8}})

Simplify

= - \frac{1}{7 x ^{7}} + \frac{1}{2 x ^{8}}

Add a constant to the solution

= - \frac{1}{7 x ^{7}} + \frac{1}{2 x ^{8}} + C

What is the integral of (x-4)/(x^9) ?
The integral of (x-4)/(x^9) is -1/(7x^7)+1/(2x^8)+C