What is the integral of y(a-by^2) ?

The integral of y(a-by^2) is 1/2 y^2(a-by^2)+(by^4)/4+C

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integral of y(a-by^2)

\int y (a - b y^{2}) d y

\frac{1}{2} y^{2} (a - b y^{2}) + \frac{b y ^{4}}{4} + C

Solution steps

\int y (a - b y^{2}) d y

Apply Integration By Parts

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

\int - b y^{3} d y = - \frac{b y ^{4}}{4}

= \frac{1}{2} y^{2} (a - b y^{2}) - (- \frac{b y ^{4}}{4})

Simplify

= \frac{1}{2} y^{2} (a - b y^{2}) + \frac{b y ^{4}}{4}

Add a constant to the solution

= \frac{1}{2} y^{2} (a - b y^{2}) + \frac{b y ^{4}}{4} + C

What is the integral of y(a-by^2) ?
The integral of y(a-by^2) is 1/2 y^2(a-by^2)+(by^4)/4+C