What is the integral of (sqrt(x)-x+5)/(x^4) ?

The integral of (sqrt(x)-x+5)/(x^4) is -2/(5x^{5/2)}+1/(2x^2)-5/(3x^3)+C

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integral of (sqrt(x)-x+5)/(x^4)

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

- \frac{2}{5 x ^{\frac{5}{2}}} + \frac{1}{2 x ^{2}} - \frac{5}{3 x ^{3}} + C

Solution steps

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

Expand

\frac{x - x + 5}{x ^{4}} : \frac{1}{x ^{\frac{7}{2}}} - \frac{1}{x ^{3}} + \frac{5}{x ^{4}}

= \int \frac{1}{x ^{\frac{7}{2}}} - \frac{1}{x ^{3}} + \frac{5}{x ^{4}} 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 ^{\frac{7}{2}}} d x - \int \frac{1}{x ^{3}} d x + \int \frac{5}{x ^{4}} d x

\int \frac{1}{x ^{\frac{7}{2}}} d x = - \frac{2}{5 x ^{\frac{5}{2}}}

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

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

= - \frac{2}{5 x ^{\frac{5}{2}}} - (- \frac{1}{2 x ^{2}}) - \frac{5}{3 x ^{3}}

Simplify

= - \frac{2}{5 x ^{\frac{5}{2}}} + \frac{1}{2 x ^{2}} - \frac{5}{3 x ^{3}}

Add a constant to the solution

= - \frac{2}{5 x ^{\frac{5}{2}}} + \frac{1}{2 x ^{2}} - \frac{5}{3 x ^{3}} + C

What is the integral of (sqrt(x)-x+5)/(x^4) ?
The integral of (sqrt(x)-x+5)/(x^4) is -2/(5x^{5/2)}+1/(2x^2)-5/(3x^3)+C