What is the integral of (2x^3-4sqrt(x))/(sqrt(x^5)) ?

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

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integral of (2x^3-4sqrt(x))/(sqrt(x^5))

\int \frac{2 x ^{3} - 4 x}{x ^{5}} d x

\frac{4}{3} x^{\frac{3}{2}} + \frac{4}{x} + C

Solution steps

\int \frac{2 x ^{3} - 4 x}{x ^{5}} d x

Expand

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

= \int \frac{2 x ^{\frac{5}{2}}}{x ^{4}} - \frac{4}{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{2 x ^{\frac{5}{2}}}{x ^{4}} d x - \int \frac{4}{x ^{4}} d x

\int \frac{2 x ^{\frac{5}{2}}}{x ^{4}} d x = \frac{4}{3} x^{\frac{3}{2}}

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

= \frac{4}{3} x^{\frac{3}{2}} - (- \frac{4}{x})

Simplify

= \frac{4}{3} x^{\frac{3}{2}} + \frac{4}{x}

Add a constant to the solution

= \frac{4}{3} x^{\frac{3}{2}} + \frac{4}{x} + C

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