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

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

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integral of (3x-5)/(-x^5)

\int \frac{3 x - 5}{- x ^{5}} d x

\frac{1}{x ^{3}} - \frac{5}{4 x ^{4}} + C

Solution steps

\int \frac{3 x - 5}{- x ^{5}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{3 x - 5}{x ^{5}} d x

Expand

\frac{3 x - 5}{x ^{5}} : \frac{3}{x ^{4}} - \frac{5}{x ^{5}}

= - \int \frac{3}{x ^{4}} - \frac{5}{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{3}{x ^{4}} d x - \int \frac{5}{x ^{5}} d x)

\int \frac{3}{x ^{4}} d x = - \frac{1}{x ^{3}}

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

= - (- \frac{1}{x ^{3}} - (- \frac{5}{4 x ^{4}}))

Simplify

= \frac{1}{x ^{3}} - \frac{5}{4 x ^{4}}

Add a constant to the solution

= \frac{1}{x ^{3}} - \frac{5}{4 x ^{4}} + C

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