What is the integral of (x-3)/(\sqrt[3]{x^5)} ?

The integral of (x-3)/(\sqrt[3]{x^5)} is 3\sqrt[3]{x}+9/(2x^{2/3)}+C

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integral of (x-3)/(\sqrt[3]{x^5)}

Solution steps

assuming

x \geq 0

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

Expand

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

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

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

\int \frac{3}{x ^{\frac{5}{3}}} d x = - \frac{9}{2 x ^{\frac{2}{3}}}

= 3 x^{\frac{1}{3}} - (- \frac{9}{2 x ^{\frac{2}{3}}})

Simplify

Add a constant to the solution

\int tan^{\frac{3}{2}} (x) sec^{4} (x) d x

\int \frac{3 x + 16}{x ^{2} - x - 6} d x

x \to 1 lim (0.8 x + 1.2)

x \to 0 lim (cos (x) sin (x))

y^{^{'}} = 2 y + 3

What is the integral of (x-3)/(\sqrt[3]{x^5)} ?
The integral of (x-3)/(\sqrt[3]{x^5)} is 3\sqrt[3]{x}+9/(2x^{2/3)}+C