Solve ∫ (from 0 to 1 - z - y) of (x+y+z) wrt x

$$\int _ { 0 } ^ { 1 - z - y } ( x + y + z ) d x$$

Evaluate

$$\frac{-\left(y+z\right)^{2}+1}{2}$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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

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

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

$$\frac{x^{2}}{2}+yx+\int z\mathrm{d}x$$

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

$$\frac{x^{2}}{2}+yx+zx$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{2}\left(1-z-y\right)^{2}+y\left(1-z-y\right)+z\left(1-z-y\right)-\left(\frac{0^{2}}{2}+y\times 0+z\times 0\right)$$

Simplify.

$$\frac{1-\left(z+y\right)^{2}}{2}$$

Differentiate w.r.t. y

$$-y-z$$