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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

Solution steps

x^{5} = x^{\frac{5}{2}},

assuming

x \geq 0

Expand

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

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

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

= 3 x^{\frac{1}{3}} + \frac{2}{5} x^{\frac{5}{2}}

Apply radical rule:

Add a constant to the solution

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