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

The integral of ((sqrt(x)+\sqrt[3]{x})^3)/x is 2/3 x^{3/2}+9/4 x^{4/3}+18/7 x^{7/6}+x+C

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

\frac{2}{3} x^{\frac{3}{2}} + \frac{9}{4} x^{\frac{4}{3}} + \frac{18}{7} x^{\frac{7}{6}} + x + C

Solution steps

Expand

= \int x^{\frac{1}{2}} + 3 x^{\frac{1}{3}} + 3 x^{\frac{1}{6}} + 1 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 x^{\frac{1}{2}} d x + \int 3 x^{\frac{1}{3}} d x + \int 3 x^{\frac{1}{6}} d x + \int 1 d x

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

\int 3 x^{\frac{1}{3}} d x = \frac{9}{4} x^{\frac{4}{3}}

\int 3 x^{\frac{1}{6}} d x = \frac{18}{7} x^{\frac{7}{6}}

\int 1 d x = x

= \frac{2}{3} x^{\frac{3}{2}} + \frac{9}{4} x^{\frac{4}{3}} + \frac{18}{7} x^{\frac{7}{6}} + x

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + \frac{9}{4} x^{\frac{4}{3}} + \frac{18}{7} x^{\frac{7}{6}} + x + C

\int \frac{e ^{t}}{1 - e ^{2 t}} d t

n = 0 \sum \infty (- 1)^{n} (n + 1)

\int x e^{(- x)} d x

f (x) = e^{l n (x)} - e^{5 l n (x^{2})}

f (x) = tan (x) cot (x)

What is the integral of ((sqrt(x)+\sqrt[3]{x})^3)/x ?
The integral of ((sqrt(x)+\sqrt[3]{x})^3)/x is 2/3 x^{3/2}+9/4 x^{4/3}+18/7 x^{7/6}+x+C