Solve ∫ (x+y+z) wrt z

$$\int ( x + y + z ) d z$$

Evaluate

$$xz+yz+\frac{z^{2}}{2}+С$$

Solution Steps

Integrate the sum term by term.

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

Find the integral of x using the table of common integrals rule \int a\mathrm{d}z=az.

$$xz+\int y\mathrm{d}z+\int z\mathrm{d}z$$

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

$$xz+yz+\int z\mathrm{d}z$$

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}.

$$xz+yz+\frac{z^{2}}{2}$$

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.

$$xz+yz+\frac{z^{2}}{2}+С$$

Differentiate w.r.t. x

$$z$$