Solve ∫ (2-6sqrt{y}) wrt y

$$\int ( 2 - 6 \sqrt { y } ) d y$$

Evaluate

$$С+2y-4y^{\frac{3}{2}}$$

Solution Steps

Integrate the sum term by term.

$$\int 2\mathrm{d}y+\int -6\sqrt{y}\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 2\mathrm{d}y-6\int \sqrt{y}\mathrm{d}y$$

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

$$2y-6\int \sqrt{y}\mathrm{d}y$$

Rewrite \sqrt{y} as y^{\frac{1}{2}}. Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{\frac{1}{2}}\mathrm{d}y with \frac{y^{\frac{3}{2}}}{\frac{3}{2}}. Simplify. Multiply -6 times \frac{2y^{\frac{3}{2}}}{3}.

$$2y-4y^{\frac{3}{2}}$$

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.

$$2y-4y^{\frac{3}{2}}+С$$

Differentiate w.r.t. y

$$-6\sqrt{y}+2$$