Solve ∫ (from 4 to 50) of (x+20^2)

$$\int _ { 4 } ^ { 50 } ( x + 20 ^ { 2 } )$$

Evaluate

$$19642$$

Solution Steps

Calculate 20 to the power of 2 and get 400.

$$\int _{4}^{50}x+400\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int x+400\mathrm{d}x$$

Integrate the sum term by term.

$$\int x\mathrm{d}x+\int 400\mathrm{d}x$$

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}.

$$\frac{x^{2}}{2}+\int 400\mathrm{d}x$$

Find the integral of 400 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{2}}{2}+400x$$

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.

$$\frac{50^{2}}{2}+400\times 50-\left(\frac{4^{2}}{2}+400\times 4\right)$$

Simplify.

$$19642$$