What is the integral of (y+2)\sqrt[3]{2-3y} ?

The integral of (y+2)\sqrt[3]{2-3y} is -1/21 (-(-3y+2)^{7/3}+14(-3y+2)^{4/3})+C

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integral of (y+2)\sqrt[3]{2-3y}

- \frac{1}{21} (- (- 3 y + 2)^{\frac{7}{3}} + 14 (- 3 y + 2)^{\frac{4}{3}}) + C

Solution steps

Apply u-substitution

Take the constant out:

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

Expand

Apply the Sum Rule:

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

\int u^{\frac{4}{3}} d u = \frac{3}{7} u^{\frac{7}{3}}

= - \frac{1}{9} (- \frac{3}{7} u^{\frac{7}{3}} + 6 u^{\frac{4}{3}})

Substitute back

u = 2 - 3 y

= - \frac{1}{9} (- \frac{3}{7} (2 - 3 y)^{\frac{7}{3}} + 6 (2 - 3 y)^{\frac{4}{3}})

Simplify

- \frac{1}{9} (- \frac{3}{7} (2 - 3 y)^{\frac{7}{3}} + 6 (2 - 3 y)^{\frac{4}{3}}) : - \frac{1}{21} (- (- 3 y + 2)^{\frac{7}{3}} + 14 (- 3 y + 2)^{\frac{4}{3}})

= - \frac{1}{21} (- (- 3 y + 2)^{\frac{7}{3}} + 14 (- 3 y + 2)^{\frac{4}{3}})

Add a constant to the solution

= - \frac{1}{21} (- (- 3 y + 2)^{\frac{7}{3}} + 14 (- 3 y + 2)^{\frac{4}{3}}) + C

What is the integral of (y+2)\sqrt[3]{2-3y} ?
The integral of (y+2)\sqrt[3]{2-3y} is -1/21 (-(-3y+2)^{7/3}+14(-3y+2)^{4/3})+C