Solve ∫ (from - 3.5 to 1) of (-2x^2-5x+7) wrt x

$$\int _ { - 3.5 } ^ { 1 } ( - 2 x ^ { 2 } - 5 x + 7 ) d x$$

Evaluate

$$30.375$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -2x^{2}-5x+7\mathrm{d}x$$

Integrate the sum term by term.

$$\int -2x^{2}\mathrm{d}x+\int -5x\mathrm{d}x+\int 7\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-2\int x^{2}\mathrm{d}x-5\int x\mathrm{d}x+\int 7\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -2 times \frac{x^{3}}{3}.

$$-\frac{2x^{3}}{3}-5\int x\mathrm{d}x+\int 7\mathrm{d}x$$

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}.

$$-\frac{2x^{3}}{3}-\frac{5x^{2}}{2}+\int 7\mathrm{d}x$$

Find the integral of 7 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{2x^{3}}{3}-\frac{5x^{2}}{2}+7x$$

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.

$$-\frac{2}{3}\times 1^{3}-\frac{5}{2}\times 1^{2}+7\times 1-\left(-\frac{2}{3}\left(-3.5\right)^{3}-\frac{5}{2}\left(-3.5\right)^{2}+7\left(-3.5\right)\right)$$

Simplify.

$$\frac{243}{8}$$