What is the integral of (x-1)^2sqrt(x) ?

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

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integral of (x-1)^2sqrt(x)

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Add a constant to the solution

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

\frac{64}{( 1 + 4 x ) ^{3}}

(x + 1) \frac{d y}{d x} - x y = x + x^{2}

x \to 9 lim (8 x + ∣ x - 9 ∣)

\frac{\partial}{\partial y} (2 x y^{3} - 3)

4 x^{\frac{5}{4}} + 8 x^{\frac{3}{2}} + 3 x

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