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

y^{^{'}} = e^{8 x} - 6 x

f (x) = \frac{40 t}{t + 2}

h \to 0 lim ((2 x + h) - 2 x)

cos (\frac{1 - e ^{4 x}}{1 + e ^{4 x}})

\int sin (x^{2} - 5) x 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