Solve ∫ y^2(y^2-3) wrt y

$$\int y ^ { 2 } ( y ^ { 2 } - 3 ) d y$$

Evaluate

$$\frac{y^{5}}{5}-y^{3}+С$$

Solution Steps

Use the distributive property to multiply y^{2} by y^{2}-3.

$$\int y^{4}-3y^{2}\mathrm{d}y$$

Integrate the sum term by term.

$$\int y^{4}\mathrm{d}y+\int -3y^{2}\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int y^{4}\mathrm{d}y-3\int y^{2}\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{4}\mathrm{d}y with \frac{y^{5}}{5}.

$$\frac{y^{5}}{5}-3\int y^{2}\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{2}\mathrm{d}y with \frac{y^{3}}{3}. Multiply -3 times \frac{y^{3}}{3}.

$$\frac{y^{5}}{5}-y^{3}$$

If F\left(y\right) is an antiderivative of f\left(y\right), then the set of all antiderivatives of f\left(y\right) is given by F\left(y\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{y^{5}}{5}-y^{3}+С$$

Differentiate w.r.t. y

$$y^{2}\left(y^{2}-3\right)$$