Solve ∫ x^3+2y wrt y

$$\int{ { x }^{ 3 } +2y }d y$$

Evaluate

$$y^{2}+yx^{3}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Find the integral of x^{3} using the table of common integrals rule \int a\mathrm{d}y=ay.

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

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}.

$$x^{3}y+y^{2}$$

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.

$$x^{3}y+y^{2}+С$$

Differentiate w.r.t. x

$$3yx^{2}$$