What is the integral of (2x+3x^{1/2})x^{1/3} ?

The integral of (2x+3x^{1/2})x^{1/3} is 6/7 x^{7/3}+18/11 x^{11/6}+C

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integral of (2x+3x^{1/2})x^{1/3}

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

\frac{6}{7} x^{\frac{7}{3}} + \frac{18}{11} x^{\frac{11}{6}} + C

Solution steps

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

Expand

(2 x + 3 x^{\frac{1}{2}}) x^{\frac{1}{3}} : 2 x^{\frac{4}{3}} + 3 x^{\frac{5}{6}}

= \int 2 x^{\frac{4}{3}} + 3 x^{\frac{5}{6}} 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 2 x^{\frac{4}{3}} d x + \int 3 x^{\frac{5}{6}} d x

\int 2 x^{\frac{4}{3}} d x = \frac{6}{7} x^{\frac{7}{3}}

\int 3 x^{\frac{5}{6}} d x = \frac{18}{11} x^{\frac{11}{6}}

= \frac{6}{7} x^{\frac{7}{3}} + \frac{18}{11} x^{\frac{11}{6}}

Add a constant to the solution

= \frac{6}{7} x^{\frac{7}{3}} + \frac{18}{11} x^{\frac{11}{6}} + C

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\frac{1}{( s + 1 ) ( ( s + 2 ) ^{2} + 2 )}

\frac{d}{d x} (5 x - sin^{2} (4 x))

\frac{d y}{d x} = 1 + x + y + x y

What is the integral of (2x+3x^{1/2})x^{1/3} ?
The integral of (2x+3x^{1/2})x^{1/3} is 6/7 x^{7/3}+18/11 x^{11/6}+C