Solve ∫ (from - 4 to 9) of a-x

$$\int _ { - 4 } ^ { 9 } a - x$$

Evaluate

$$13a-\frac{65}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int a-x\mathrm{d}x$$

Integrate the sum term by term.

$$\int a\mathrm{d}x+\int -x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int a\mathrm{d}x-\int x\mathrm{d}x$$

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

$$ax-\int x\mathrm{d}x$$

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 -1 times \frac{x^{2}}{2}.

$$ax-\frac{x^{2}}{2}$$

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.

$$a\times 9-\frac{9^{2}}{2}-\left(a\left(-4\right)-\frac{\left(-4\right)^{2}}{2}\right)$$

Simplify.

$$-\frac{65}{2}+13a$$

Differentiate w.r.t. a

$$13$$