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

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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

u^{4}

f (x) = (1 - x) (x^{2} - 3)^{2}, at x = 2

x \to 2 + lim ((2 - x)^{x - 2})

3 y^{^{''}} - 5 y^{^{'}} - 14 y = 0

\int_{4}^{7} (x - 6)^{2} d x

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