Solve ∫ 2(y-1) wrt y

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

Evaluate

$$y^{2}-2y+С$$

Solution Steps

Use the distributive property to multiply 2 by y-1.

$$\int 2y-2\mathrm{d}y$$

Integrate the sum term by term.

$$\int 2y\mathrm{d}y+\int -2\mathrm{d}y$$

Factor out the constant in each of the terms.

$$2\int y\mathrm{d}y+\int -2\mathrm{d}y$$

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

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

Find the integral of -2 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$y^{2}-2y$$

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.

$$y^{2}-2y+С$$

Differentiate w.r.t. y

$$2\left(y-1\right)$$