Solve ∫ (from - 5 to 4) of 4x-5x

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

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

$$\int 4x-5x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4x\mathrm{d}x+\int -5x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$4\int x\mathrm{d}x-5\int x\mathrm{d}x$$

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 4 times \frac{x^{2}}{2}.

$$2x^{2}-5\int x\mathrm{d}x$$

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 -5 times \frac{x^{2}}{2}.

$$2x^{2}-\frac{5x^{2}}{2}$$

Simplify.

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

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.

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

Simplify.

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