What is the integral of (x\sqrt[4]{x^3}+sqrt(x))/(x^2) ?

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

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integral of (x\sqrt[4]{x^3}+sqrt(x))/(x^2)

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

Solution steps

assuming

x \geq 0

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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