Solve ∫ (from - 1 to 0) of (2x^3-5x) wrt x

$$\int _ { - 1 } ^ { 0 } ( 2 x ^ { 3 } - 5 x ) d x$$

Evaluate

$$2$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 2 times \frac{x^{4}}{4}.

$$\frac{x^{4}}{2}-5\int x\mathrm{d}x$$

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 -5 times \frac{x^{2}}{2}.

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

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.

$$\frac{0^{4}}{2}-\frac{5}{2}\times 0^{2}-\left(\frac{\left(-1\right)^{4}}{2}-\frac{5}{2}\left(-1\right)^{2}\right)$$

Simplify.

$$2$$