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

The integral of (2x-x^{1/2})^3 is 2x^4-24/7 x^{7/2}+2x^3-2/5 x^{5/2}+C

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

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

2 x^{4} - \frac{24}{7} x^{\frac{7}{2}} + 2 x^{3} - \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

(2 x - x^{\frac{1}{2}})^{3} : 8 x^{3} - 12 x^{\frac{5}{2}} + 6 x^{2} - x^{\frac{3}{2}}

= \int 8 x^{3} - 12 x^{\frac{5}{2}} + 6 x^{2} - 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 8 x^{3} d x - \int 12 x^{\frac{5}{2}} d x + \int 6 x^{2} d x - \int x^{\frac{3}{2}} d x

\int 8 x^{3} d x = 2 x^{4}

\int 12 x^{\frac{5}{2}} d x = \frac{24}{7} x^{\frac{7}{2}}

\int 6 x^{2} d x = 2 x^{3}

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

= 2 x^{4} - \frac{24}{7} x^{\frac{7}{2}} + 2 x^{3} - \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= 2 x^{4} - \frac{24}{7} x^{\frac{7}{2}} + 2 x^{3} - \frac{2}{5} x^{\frac{5}{2}} + C

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What is the integral of (2x-x^{1/2})^3 ?
The integral of (2x-x^{1/2})^3 is 2x^4-24/7 x^{7/2}+2x^3-2/5 x^{5/2}+C