What is the integral of (sqrt(x)-2x^2)/(x^3) ?

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

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integral of (sqrt(x)-2x^2)/(x^3)

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

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

Solution steps

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

Expand

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

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

Add a constant to the solution

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

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