Solve ∫ (7-3y^2)^1 wrt y

$$\int ( 7 - 3 y ^ { 2 } ) ^ { 1 } d y$$

Evaluate

$$С+7y-y^{3}$$

Solution Steps

Calculate 7-3y^{2} to the power of 1 and get 7-3y^{2}.

$$\int 7-3y^{2}\mathrm{d}y$$

Integrate the sum term by term.

$$\int 7\mathrm{d}y+\int -3y^{2}\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 7\mathrm{d}y-3\int y^{2}\mathrm{d}y$$

Find the integral of 7 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$7y-3\int y^{2}\mathrm{d}y$$

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}.

$$7y-y^{3}$$

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.

$$7y-y^{3}+С$$

Differentiate w.r.t. y

$$7-3y^{2}$$