Solve ∫ (from - 10 to 15) of (-11x^2-11x^4) wrt x

$$\int _ { - 10 } ^ { 15 } ( - 11 x ^ { 2 } - 11 x ^ { 4 } ) d x$$

Evaluate

$$-\frac{5720000}{3}\approx -1906666.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -11x^{2}-11x^{4}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -11x^{2}\mathrm{d}x+\int -11x^{4}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-11\int x^{2}\mathrm{d}x-11\int x^{4}\mathrm{d}x$$

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 -11 times \frac{x^{3}}{3}.

$$-\frac{11x^{3}}{3}-11\int x^{4}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply -11 times \frac{x^{5}}{5}.

$$-\frac{11x^{3}}{3}-\frac{11x^{5}}{5}$$

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.

$$-\frac{11}{3}\times 15^{3}-\frac{11}{5}\times 15^{5}-\left(-\frac{11}{3}\left(-10\right)^{3}-\frac{11}{5}\left(-10\right)^{5}\right)$$

Simplify.

$$-\frac{5720000}{3}$$