What is the integral of 2sqrt(y)(-y+2) ?

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

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integral of 2sqrt(y)(-y+2)

\int 2 y (- y + 2) d y

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

Solution steps

\int 2 y (- y + 2) d y

Take the constant out:

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

= 2 \cdot \int y (- y + 2) d y

Expand

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

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

Apply the Sum Rule:

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

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

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

\int 2 y d y = \frac{4}{3} y^{\frac{3}{2}}

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

Add a constant to the solution

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

\frac{d}{d x} (ln (x + 1) - x + 1)

x \to 5 lim (4 x^{2} - 8)

\frac{d}{d x} (x x + 9)

\frac{9}{4} t^{2} e^{5 t}

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