Solve ∫ int{int{x wrt xy}dyz}dz

$$\int{ \int{ \int{ x }d x y }d y z }d z$$

Evaluate

$$\frac{С\left(yz\right)^{2}}{4}+\frac{\left(xyz\right)^{2}}{8}+\frac{С_{1}z^{2}}{2}+С_{2}$$

Solution Steps

Factor out the constant using \int af\left(z\right)\mathrm{d}z=a\int f\left(z\right)\mathrm{d}z.

$$\int \int x\mathrm{d}xy\mathrm{d}y\int z\mathrm{d}z$$

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

$$\int \int x\mathrm{d}xy\mathrm{d}y\times \frac{z^{2}}{2}$$

Simplify.

$$\frac{\left(\frac{\left(\frac{x^{2}}{2}+С\right)y^{2}}{2}+С\right)z^{2}}{2}$$

If F\left(z\right) is an antiderivative of f\left(z\right), then the set of all antiderivatives of f\left(z\right) is given by F\left(z\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{\left(\frac{\left(\frac{x^{2}}{2}+С\right)y^{2}}{2}+С\right)z^{2}}{2}+С$$

Differentiate w.r.t. x

$$\frac{x\left(yz\right)^{2}}{4}$$