Solve ∫ (from 0 to 2) of -x^2+x-3x^3-x^2-11x wrt x

$$\int_{ 0 }^{ 2 } - { x }^{ 2 } +x-3 { x }^{ 3 } - { x }^{ 2 } -11x d x$$

Evaluate

$$-\frac{112}{3}\approx -37.333333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -x^{2}+x-3x^{3}-x^{2}-11x\mathrm{d}x$$

Integrate the sum term by term.

$$\int -x^{2}\mathrm{d}x+\int x\mathrm{d}x+\int -3x^{3}\mathrm{d}x+\int -x^{2}\mathrm{d}x+\int -11x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\int x^{2}\mathrm{d}x+\int x\mathrm{d}x-3\int x^{3}\mathrm{d}x-\int x^{2}\mathrm{d}x-11\int x\mathrm{d}x$$

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

$$-\frac{x^{3}}{3}+\int x\mathrm{d}x-3\int x^{3}\mathrm{d}x-\int x^{2}\mathrm{d}x-11\int x\mathrm{d}x$$

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}.

$$-\frac{x^{3}}{3}+\frac{x^{2}}{2}-3\int x^{3}\mathrm{d}x-\int x^{2}\mathrm{d}x-11\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply -3 times \frac{x^{4}}{4}.

$$-\frac{x^{3}}{3}+\frac{x^{2}}{2}-\frac{3x^{4}}{4}-\int x^{2}\mathrm{d}x-11\int x\mathrm{d}x$$

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

$$-\frac{x^{3}}{3}+\frac{x^{2}}{2}-\frac{3x^{4}}{4}-\frac{x^{3}}{3}-11\int x\mathrm{d}x$$

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 -11 times \frac{x^{2}}{2}.

$$-\frac{x^{3}}{3}+\frac{x^{2}}{2}-\frac{3x^{4}}{4}-\frac{x^{3}}{3}-\frac{11x^{2}}{2}$$

Simplify.

$$-\frac{2x^{3}}{3}-5x^{2}-\frac{3x^{4}}{4}$$

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.

$$-\frac{2}{3}\times 2^{3}-5\times 2^{2}-\frac{3}{4}\times 2^{4}-\left(-\frac{2}{3}\times 0^{3}-5\times 0^{2}-\frac{3}{4}\times 0^{4}\right)$$

Simplify.

$$-\frac{112}{3}$$