What is the integral of \sqrt[3]{by^2} ?

The integral of \sqrt[3]{by^2} is y\sqrt[3]{by^2}-(2\sqrt[3]{b}y^{5/3})/5+C

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integral of \sqrt[3]{by^2}

\int 3 b y^{2} d y

y 3 b y^{2} - \frac{2 3 b y ^{\frac{5}{3}}}{5} + C

Solution steps

\int 3 b y^{2} d y

Apply Integration By Parts

: y 3 b y^{2} - \int \frac{2 b ^{\frac{1}{3}} y ^{2}}{3 ( y ^{2} ) ^{\frac{2}{3}}} d y

= y 3 b y^{2} - \int \frac{2 b ^{\frac{1}{3}} y ^{2}}{3 ( y ^{2} ) ^{\frac{2}{3}}} d y

\int \frac{2 b ^{\frac{1}{3}} y ^{2}}{3 ( y ^{2} ) ^{\frac{2}{3}}} d y = \frac{2 b ^{\frac{1}{3}} y ^{\frac{5}{3}}}{5}

= y 3 b y^{2} - \frac{2 b ^{\frac{1}{3}} y ^{\frac{5}{3}}}{5}

Apply radical rule:

a^{\frac{1}{n}} = n a

= y 3 b y^{2} - \frac{2 3 b y ^{\frac{5}{3}}}{5}

Add a constant to the solution

= y 3 b y^{2} - \frac{2 3 b y ^{\frac{5}{3}}}{5} + C

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

\int x^{2} 3 + 7 x^{2} d x

\frac{d}{d x} (ln (y - x^{2}))

\int x \cdot e^{x^{2} + 1} d x

(5 x + y e^{\frac{y}{x}}) d x - (x e^{\frac{y}{x}}) d y = 0, y (1) = 3

What is the integral of \sqrt[3]{by^2} ?
The integral of \sqrt[3]{by^2} is y\sqrt[3]{by^2}-(2\sqrt[3]{b}y^{5/3})/5+C