What is the integral of x^5(2-x^3)^{1/2} ?

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

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integral of x^5(2-x^3)^{1/2}

integral

x^{5} (2 - x^{3})^{\frac{1}{2}}

- \frac{1}{3} (- \frac{2}{5} (2 - x^{3})^{\frac{5}{2}} + \frac{4}{3} (2 - x^{3})^{\frac{3}{2}}) + C

Solution steps

\int x^{5} (2 - x^{3})^{\frac{1}{2}} d x

Apply u-substitution

: \int - \frac{( - u + 2 ) u ^{\frac{1}{2}}}{3} d u

= \int - \frac{( - u + 2 ) u ^{\frac{1}{2}}}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{3} \cdot \int (- u + 2) u^{\frac{1}{2}} d u

Expand

(- u + 2) u^{\frac{1}{2}} : - u^{\frac{3}{2}} + 2 u^{\frac{1}{2}}

= - \frac{1}{3} \cdot \int - u^{\frac{3}{2}} + 2 u^{\frac{1}{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \frac{1}{3} (- \int u^{\frac{3}{2}} d u + \int 2 u^{\frac{1}{2}} d u)

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

\int 2 u^{\frac{1}{2}} d u = \frac{4}{3} u^{\frac{3}{2}}

= - \frac{1}{3} (- \frac{2}{5} u^{\frac{5}{2}} + \frac{4}{3} u^{\frac{3}{2}})

Substitute back

u = 2 - x^{3}

= - \frac{1}{3} (- \frac{2}{5} (2 - x^{3})^{\frac{5}{2}} + \frac{4}{3} (2 - x^{3})^{\frac{3}{2}})

Add a constant to the solution

= - \frac{1}{3} (- \frac{2}{5} (2 - x^{3})^{\frac{5}{2}} + \frac{4}{3} (2 - x^{3})^{\frac{3}{2}}) + C

9 x^{2} sin (4 x)

\frac{1}{( e ^{t} - 2 )}

sec (3 x + 2) tan (3 x + 2)

x^{7} - 2 x^{5} + 2 x^{5} + x - 1

x^{\frac{1}{8}}

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