Solve ∫ (x^2+y^2) wrt y

$$\int ( x ^ { 2 } + y ^ { 2 } ) d y$$

Evaluate

$$yx^{2}+\frac{y^{3}}{3}+С$$

Solution Steps

Integrate the sum term by term.

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

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

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

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}.

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

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.

$$x^{2}y+\frac{y^{3}}{3}+С$$

Differentiate w.r.t. x

$$2xy$$