What is the integral of (8x-x^{5/2}-2sqrt(x))/(x^2) ?

The integral of (8x-x^{5/2}-2sqrt(x))/(x^2) is 8ln|x|-2/3 x^{3/2}+4/(sqrt(x))+C

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integral of (8x-x^{5/2}-2sqrt(x))/(x^2)

\int \frac{8 x - x ^{\frac{5}{2}} - 2 x}{x ^{2}} d x

8 ln ∣ x ∣ - \frac{2}{3} x^{\frac{3}{2}} + \frac{4}{x} + C

Solution steps

\int \frac{8 x - x ^{\frac{5}{2}} - 2 x}{x ^{2}} d x

Expand

\frac{8 x - x ^{\frac{5}{2}} - 2 x}{x ^{2}} : \frac{8}{x} - x^{\frac{1}{2}} - \frac{2}{x ^{\frac{3}{2}}}

= \int \frac{8}{x} - x^{\frac{1}{2}} - \frac{2}{x ^{\frac{3}{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{8}{x} d x - \int x^{\frac{1}{2}} d x - \int \frac{2}{x ^{\frac{3}{2}}} d x

\int \frac{8}{x} d x = 8 ln ∣ x ∣

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

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

= 8 ln ∣ x ∣ - \frac{2}{3} x^{\frac{3}{2}} - (- \frac{4}{x ^{\frac{1}{2}}})

Simplify

8 ln ∣ x ∣ - \frac{2}{3} x^{\frac{3}{2}} - (- \frac{4}{x ^{\frac{1}{2}}}) : 8 ln ∣ x ∣ - \frac{2}{3} x^{\frac{3}{2}} + \frac{4}{x}

= 8 ln ∣ x ∣ - \frac{2}{3} x^{\frac{3}{2}} + \frac{4}{x}

Add a constant to the solution

= 8 ln ∣ x ∣ - \frac{2}{3} x^{\frac{3}{2}} + \frac{4}{x} + C

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What is the integral of (8x-x^{5/2}-2sqrt(x))/(x^2) ?
The integral of (8x-x^{5/2}-2sqrt(x))/(x^2) is 8ln|x|-2/3 x^{3/2}+4/(sqrt(x))+C