Solve ∫ (from 4 to 5) of ax

$$\int _ { 4 } ^ { 5 } a x$$

Evaluate

$$\frac{9a}{2}$$

Solution Steps

Evaluate the indefinite integral first.

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

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

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

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}.

$$a\times \frac{x^{2}}{2}$$

Simplify.

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

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.

$$\frac{1}{2}a\times 5^{2}-\frac{1}{2}a\times 4^{2}$$

Simplify.

$$\frac{9a}{2}$$

Differentiate w.r.t. a

$$\frac{9}{2} = 4\frac{1}{2} = 4.5$$