What is the integral of y(b-ay^2) ?

The integral of y(b-ay^2) is 1/2 y^2(b-ay^2)+(ay^4)/4+C

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integral of y(b-ay^2)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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