Solve ∫ (from 5 to 10) of (8x+5) wrt x

$$\int _ { 5 } ^ { 10 } ( 8 x + 5 ) d x$$

Evaluate

$$325$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 8x+5\mathrm{d}x$$

Integrate the sum term by term.

$$\int 8x\mathrm{d}x+\int 5\mathrm{d}x$$

Factor out the constant in each of the terms.

$$8\int x\mathrm{d}x+\int 5\mathrm{d}x$$

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}. Multiply 8 times \frac{x^{2}}{2}.

$$4x^{2}+\int 5\mathrm{d}x$$

Find the integral of 5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$4x^{2}+5x$$

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.

$$4\times 10^{2}+5\times 10-\left(4\times 5^{2}+5\times 5\right)$$

Simplify.

$$325$$