Solve ∫ (from - 4 to 1) of (5x^2-80) wrt x

$$\int _ { - 4 } ^ { 1 } ( 5 x ^ { 2 } - 80 ) d x$$

Evaluate

$$-\frac{875}{3}\approx -291.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5x^{2}-80\mathrm{d}x$$

Integrate the sum term by term.

$$\int 5x^{2}\mathrm{d}x+\int -80\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int x^{2}\mathrm{d}x+\int -80\mathrm{d}x$$

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 5 times \frac{x^{3}}{3}.

$$\frac{5x^{3}}{3}+\int -80\mathrm{d}x$$

Find the integral of -80 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{5x^{3}}{3}-80x$$

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.

$$\frac{5}{3}\times 1^{3}-80-\left(\frac{5}{3}\left(-4\right)^{3}-80\left(-4\right)\right)$$

Simplify.

$$-\frac{875}{3}$$