Solve ∫ (from 2 to 3) of (82^3+3z-1) wrt z=

$$\int _ { 2 } ^ { 3 } ( 82 ^ { 3 } + 3 z - 1 ) d z =$$

Evaluate

$$\frac{1102749}{2}=551374.5$$

Solution Steps

Calculate 82 to the power of 3 and get 551368.

$$\int _{2}^{3}551368+3z-1\mathrm{d}z$$

Subtract 1 from 551368 to get 551367.

$$\int _{2}^{3}551367+3z\mathrm{d}z$$

Evaluate the indefinite integral first.

$$\int 551367+3z\mathrm{d}z$$

Integrate the sum term by term.

$$\int 551367\mathrm{d}z+\int 3z\mathrm{d}z$$

Factor out the constant in each of the terms.

$$\int 551367\mathrm{d}z+3\int z\mathrm{d}z$$

Find the integral of 551367 using the table of common integrals rule \int a\mathrm{d}z=az.

$$551367z+3\int z\mathrm{d}z$$

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}. Multiply 3 times \frac{z^{2}}{2}.

$$551367z+\frac{3z^{2}}{2}$$

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.

$$551367\times 3+\frac{3}{2}\times 3^{2}-\left(551367\times 2+\frac{3}{2}\times 2^{2}\right)$$

Simplify.

$$\frac{1102749}{2}$$