Solve ∫ (from 12 to a + 4 * 5) of ( -0.2x+9) wrt x

$$\int _ { 12 } ^ { a + 4 \cdot 5 } ( - 0.2 x + 9 ) d x$$

Evaluate

$$-\frac{\left(a-58\right)\left(a+8\right)}{10}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -\frac{x}{5}+9\mathrm{d}x$$

Integrate the sum term by term.

$$\int -\frac{x}{5}\mathrm{d}x+\int 9\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\frac{\int x\mathrm{d}x}{5}+\int 9\mathrm{d}x$$

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 -0.2 times \frac{x^{2}}{2}.

$$-\frac{x^{2}}{10}+\int 9\mathrm{d}x$$

Find the integral of 9 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{x^{2}}{10}+9x$$

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.

$$-\frac{1}{10}\left(a+4\times 5\right)^{2}+9\left(a+4\times 5\right)-\left(-\frac{12^{2}}{10}+9\times 12\right)$$

Simplify.

$$-\frac{\left(-58+a\right)\left(8+a\right)}{10}$$

Differentiate w.r.t. a

$$-\frac{a}{5}+5$$