Solve ∫ (from - 3 to 1) of (-x^3+2x) wrt x=

$$\int _ { - 3 } ^ { 1 } ( - x ^ { 3 } + 2 x ) d x =$$

Evaluate

$$12$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 -1 times \frac{x^{4}}{4}.

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

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

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

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.

$$-\frac{1^{4}}{4}+1^{2}-\left(-\frac{\left(-3\right)^{4}}{4}+\left(-3\right)^{2}\right)$$

Simplify.

$$12$$